A Bouncing Ball Never Quite Stops, at Least on Paper
Learn to test whether a sequence is a geometric progression and state its common ratio, find any term of a GP, model doubling growth and decay, and write the sequence for a self-similar pattern.
How do you describe a quantity that keeps shrinking by the same fraction?
Drop a ball from cm. It rebounds to three quarters of that height, then three quarters of the new height, and so on.
The differences are , , — nothing constant there, so this is not an arithmetic progression. But look at the ratios:
Constant. A sequence where each term is the previous one multiplied by a fixed number is a geometric progression, and that fixed number is the common ratio .
Here is what makes a GP different from an AP in a way worth noticing. An AP with a negative difference eventually goes past zero into negative values — a temperature falling degrees an hour gets below freezing. A GP with between and gets smaller and smaller and never reaches zero. The ball's height after ten bounces is tiny but positive, and after a hundred it is still positive.
So on paper the bouncing never stops. That is the model departing from the real ball, and the last section of this page comes back to it.
This page covers the third part of the CBSE Class 9 Mathematics chapter on sequences — recognising a GP, finding its terms, growth and decay, and self-similar patterns.
The differences are , , — nothing constant there, so this is not an arithmetic progression. But look at the ratios:
Constant. A sequence where each term is the previous one multiplied by a fixed number is a geometric progression, and that fixed number is the common ratio .
Here is what makes a GP different from an AP in a way worth noticing. An AP with a negative difference eventually goes past zero into negative values — a temperature falling degrees an hour gets below freezing. A GP with between and gets smaller and smaller and never reaches zero. The ball's height after ten bounces is tiny but positive, and after a hundred it is still positive.
So on paper the bouncing never stops. That is the model departing from the real ball, and the last section of this page comes back to it.
This page covers the third part of the CBSE Class 9 Mathematics chapter on sequences — recognising a GP, finding its terms, growth and decay, and self-similar patterns.
How do you tell whether a sequence is a geometric progression?
Divide each term by the one before it. If every answer is the same, it is a GP, and that shared answer is the common ratio .
Worked example 1 — a GP. , , , . Ratios: , , . So and .
Worked example 2 — a bigger ratio. , , , . Ratios all , so and .
Worked example 3 — a shrinking GP. , , , . Ratios all , so and .
Worked example 4 — a negative ratio. , , , . Ratios: , then , then . So , and the signs alternate — which is exactly what a negative ratio does.
Worked example 5 — not a GP. , , , . Ratios: , , — not constant. These are the square numbers, and they are neither an AP nor a GP.
Worked example 6 — an AP, not a GP. , , , . Ratios: , , — not constant. But the differences are all , so it is an AP.
Worked example 7 — the pair that starts the same way. Both and begin with . The first continues by adding and the second by multiplying by . Two terms never settle which kind of progression you are looking at — a third is the minimum, and even then a fourth is worth checking.
**Worked example 8 — finding from two terms.** In a GP the second term is and the fifth is . Three multiplications separate them:
and working back one step, . The GP is
Check both the differences and the ratios before deciding. A sequence is an AP if the differences are constant and a GP if the ratios are, and it can be neither. Testing only one of the two is how the square numbers get mislabelled, and the test costs three divisions.
**Neither nor may be zero.** If every term is zero; if every term after the first is zero. In both cases there is no ratio to speak of, which is why the definition excludes them — much as was required of a linear polynomial for the same kind of reason.
Worked example 1 — a GP. , , , . Ratios: , , . So and .
Worked example 2 — a bigger ratio. , , , . Ratios all , so and .
Worked example 3 — a shrinking GP. , , , . Ratios all , so and .
Worked example 4 — a negative ratio. , , , . Ratios: , then , then . So , and the signs alternate — which is exactly what a negative ratio does.
Worked example 5 — not a GP. , , , . Ratios: , , — not constant. These are the square numbers, and they are neither an AP nor a GP.
Worked example 6 — an AP, not a GP. , , , . Ratios: , , — not constant. But the differences are all , so it is an AP.
Worked example 7 — the pair that starts the same way. Both and begin with . The first continues by adding and the second by multiplying by . Two terms never settle which kind of progression you are looking at — a third is the minimum, and even then a fourth is worth checking.
**Worked example 8 — finding from two terms.** In a GP the second term is and the fifth is . Three multiplications separate them:
and working back one step, . The GP is
Check both the differences and the ratios before deciding. A sequence is an AP if the differences are constant and a GP if the ratios are, and it can be neither. Testing only one of the two is how the square numbers get mislabelled, and the test costs three divisions.
**Neither nor may be zero.** If every term is zero; if every term after the first is zero. In both cases there is no ratio to speak of, which is why the definition excludes them — much as was required of a linear polynomial for the same kind of reason.
Formula
How do you find the nth term of a GP?
**Start at the first term and multiply by the common ratio times:**
The index is for the same reason as in an AP — reaching the fourth term takes three multiplications.
Worked example 1. For , , , with , :
Check the small cases: and . Both correct.
Worked example 2. For , , with :
Worked example 3 — a shrinking GP. For , , with :
Worked example 4 — locating a term. Which term of , , , equals ?
Check: . Correct.
Worked example 5 — a value that is not a term. Is a term of , , , ? Then , which is not even a whole number, let alone a power of . **So is not a term.
Worked example 6 — a negative ratio.** For , , with :
The fourth power of a negative number is positive, so — and the terms run , with odd-numbered terms positive and even-numbered terms negative.
Compare the two growth rules side by side. Take with a step of in both senses:
- AP with : and
- GP with : and
The GP is not a little bigger; it is in a different league. After twenty terms one has reached and the other a number over a million, and that gap is the whole reason growth problems are modelled geometrically. The same contrast in reverse is why decay problems are too.
The index is for the same reason as in an AP — reaching the fourth term takes three multiplications.
Worked example 1. For , , , with , :
Check the small cases: and . Both correct.
Worked example 2. For , , with :
Worked example 3 — a shrinking GP. For , , with :
Worked example 4 — locating a term. Which term of , , , equals ?
Check: . Correct.
Worked example 5 — a value that is not a term. Is a term of , , , ? Then , which is not even a whole number, let alone a power of . **So is not a term.
Worked example 6 — a negative ratio.** For , , with :
The fourth power of a negative number is positive, so — and the terms run , with odd-numbered terms positive and even-numbered terms negative.
Compare the two growth rules side by side. Take with a step of in both senses:
- AP with : and
- GP with : and
The GP is not a little bigger; it is in a different league. After twenty terms one has reached and the other a number over a million, and that gap is the whole reason growth problems are modelled geometrically. The same contrast in reverse is why decay problems are too.
How do you model doubling growth and decay with a GP?
**Write the starting amount as and the per-step multiplier as **, then read off the term you need. A growth factor makes ; a loss of a fixed percentage makes .
Worked example 1 — bacteria doubling. A culture starts with cells and doubles every hour. After hours the count is , so after hours:
Note the index. Here the exponent is , not , because *after hours is the starting count. *Whether the formula carries or depends on where the counting starts, so write out the first two or three values and check against the wording before trusting either.
Worked example 2 — the bouncing ball.** A ball dropped from cm rebounds to of its previous height. The heights are , , , , and after four bounces:
Check step by step: . Correct.
Worked example 3 — depreciation. A machine worth loses per cent of its value each year. Losing a tenth means keeping nine tenths, so :
**A fall of per cent a year is not a fall of per cent in three years.** The loss each year is per cent of a smaller amount, so the total fall is per cent, not . That is the difference between geometric decay and arithmetic decay, and it is the most common error in these questions.
Worked example 4 — compound growth of a deposit. A deposit of grows by per cent a year, so :
Simple interest at the same rate would give . The extra is the interest earning interest.
Worked example 5 — halving. A quantity of g halves every day. After days it is , so after days:
Worked example 6 — working out how many steps. A population of doubles every hour. When does it pass ?
Since and , the answer is hours. Check: after hours the count is , and after it is .
**A model with approaches zero without reaching it.** The ball's rebound height after bounces is , a very small positive number — and the real ball stopped bouncing long before that. The mathematics is not wrong; it is describing an idealised ball, and knowing where a model stops applying is part of using one properly.
Worked example 1 — bacteria doubling. A culture starts with cells and doubles every hour. After hours the count is , so after hours:
Note the index. Here the exponent is , not , because *after hours is the starting count. *Whether the formula carries or depends on where the counting starts, so write out the first two or three values and check against the wording before trusting either.
Worked example 2 — the bouncing ball.** A ball dropped from cm rebounds to of its previous height. The heights are , , , , and after four bounces:
Check step by step: . Correct.
Worked example 3 — depreciation. A machine worth loses per cent of its value each year. Losing a tenth means keeping nine tenths, so :
**A fall of per cent a year is not a fall of per cent in three years.** The loss each year is per cent of a smaller amount, so the total fall is per cent, not . That is the difference between geometric decay and arithmetic decay, and it is the most common error in these questions.
Worked example 4 — compound growth of a deposit. A deposit of grows by per cent a year, so :
Simple interest at the same rate would give . The extra is the interest earning interest.
Worked example 5 — halving. A quantity of g halves every day. After days it is , so after days:
Worked example 6 — working out how many steps. A population of doubles every hour. When does it pass ?
Since and , the answer is hours. Check: after hours the count is , and after it is .
**A model with approaches zero without reaching it.** The ball's rebound height after bounces is , a very small positive number — and the real ball stopped bouncing long before that. The mathematics is not wrong; it is describing an idealised ball, and knowing where a model stops applying is part of using one properly.
How do you write the sequence for a self-similar pattern?
Count how many copies each piece becomes at the next stage, and that count is the common ratio. Self-similar patterns are geometric progressions by construction.
Worked example 1 — the Sierpinski triangle. Start with one shaded triangle. At each stage, every shaded triangle is divided into four smaller ones and the middle one is removed, leaving three. The number of shaded triangles at each stage is
a GP with and . At stage , counting the start as stage :
So stage has shaded triangles.
Worked example 2 — the area that survives. Each stage keeps of the shaded area, since one of four parts is removed. The fraction remaining is
Another GP, with . After stages, of the original area remains — a little under a third.
Worked example 3 — the side length. Each new triangle has half the side of the one it came from, so the side lengths run , , , — a GP with . At stage the side is of the original.
Worked example 4 — the Sierpinski carpet. Start with a square. At each stage every shaded square splits into nine and the central one is removed, leaving eight:
a GP with , so stage has squares. Stage has .
The area remaining is , so after stages it is — about per cent.
Worked example 5 — three sequences from one picture. For the triangle at stage :
- pieces:
- side of each piece: of the original
- total area:
Check the consistency: pieces, each of side , so each has of the original area, giving in total. The three sequences agree, which is the sign that the counting was done correctly.
The count grows while the area shrinks. At stage the triangle has pieces and retains only of its area — roughly per cent. **Both are GPs, one with and one with , describing the same figure.
Counting pieces and counting area are different questions, and both are asked.** The triangle question *how many shaded parts at stage * has the answer , while what fraction of the area remains has the answer . The numerators match by coincidence of the numbers chosen, which makes this a good place to read the question twice before answering.
Worked example 1 — the Sierpinski triangle. Start with one shaded triangle. At each stage, every shaded triangle is divided into four smaller ones and the middle one is removed, leaving three. The number of shaded triangles at each stage is
a GP with and . At stage , counting the start as stage :
So stage has shaded triangles.
Worked example 2 — the area that survives. Each stage keeps of the shaded area, since one of four parts is removed. The fraction remaining is
Another GP, with . After stages, of the original area remains — a little under a third.
Worked example 3 — the side length. Each new triangle has half the side of the one it came from, so the side lengths run , , , — a GP with . At stage the side is of the original.
Worked example 4 — the Sierpinski carpet. Start with a square. At each stage every shaded square splits into nine and the central one is removed, leaving eight:
a GP with , so stage has squares. Stage has .
The area remaining is , so after stages it is — about per cent.
Worked example 5 — three sequences from one picture. For the triangle at stage :
- pieces:
- side of each piece: of the original
- total area:
Check the consistency: pieces, each of side , so each has of the original area, giving in total. The three sequences agree, which is the sign that the counting was done correctly.
The count grows while the area shrinks. At stage the triangle has pieces and retains only of its area — roughly per cent. **Both are GPs, one with and one with , describing the same figure.
Counting pieces and counting area are different questions, and both are asked.** The triangle question *how many shaded parts at stage * has the answer , while what fraction of the area remains has the answer . The numerators match by coincidence of the numbers chosen, which makes this a good place to read the question twice before answering.
Exam tip
Exam tip: test the ratios AND the differences before you decide
Divide consecutive terms to test for a GP, and subtract them to test for an AP. is neither, and testing only one way is how it gets mislabelled.
Two terms never settle it. and start the same and are different progressions.
**Write and on their own line** before substituting. For that is , .
**The th term is — the fourth term needs three** multiplications. Test at : it must return .
**A negative alternates the signs.** For , odd-numbered terms are positive and even-numbered ones negative.
**For "which term equals ", write both sides as powers**: gives . If no whole power works, the value is not a term.
**Losing per cent means multiplying by ** — a per cent loss gives , and three years of it is , not a per cent fall.
**Check whether the exponent is or ** by writing the first two values against the wording. *After hours* means ; the first term means .
Show the step-by-step values for a growth or decay question as well as the formula answer — it is the check and it earns marks.
For self-similar patterns, count what one piece becomes — three for the Sierpinski triangle, eight for the carpet.
And read whether pieces or area is wanted: triangles against of the area.
Two terms never settle it. and start the same and are different progressions.
**Write and on their own line** before substituting. For that is , .
**The th term is — the fourth term needs three** multiplications. Test at : it must return .
**A negative alternates the signs.** For , odd-numbered terms are positive and even-numbered ones negative.
**For "which term equals ", write both sides as powers**: gives . If no whole power works, the value is not a term.
**Losing per cent means multiplying by ** — a per cent loss gives , and three years of it is , not a per cent fall.
**Check whether the exponent is or ** by writing the first two values against the wording. *After hours* means ; the first term means .
Show the step-by-step values for a growth or decay question as well as the formula answer — it is the check and it earns marks.
For self-similar patterns, count what one piece becomes — three for the Sierpinski triangle, eight for the carpet.
And read whether pieces or area is wanted: triangles against of the area.
Did you know
Why halving a distance for ever still gets you across the room
Stand at one end of a room and walk half way across. Then half of what remains. Then half of that.
The distances you cover form a GP:
There are infinitely many steps, and no step ever takes you to the wall. So it can look as though the room is impossible to cross.
But add up how far you have gone. After one step, . After two, . After three, . After four, . After ten, .
Those totals are creeping up on and never passing it. The pattern in the numerators is unmistakable: each total is , one short of the whole in a denominator that doubles every time.
So the sum of infinitely many shrinking distances is a perfectly finite number. The room is one room wide, and the infinitely many steps fit inside it with nothing left over.
The same arithmetic answers the bouncing ball. Its rebound heights , , , go on for ever in the model, and the total distance travelled is finite — the ball comes to rest in a finite time having covered a finite path, even though the model gives it infinitely many bounces. Nothing contradictory is happening; the bounces get shorter faster than they get more numerous.
And the repeating decimal from the numbers chapter is the same object once more. means , a GP with , and its total is exactly . Infinitely many terms, one clean answer — which is why the sum of a GP with gets a formula of its own in Class 11.
The distances you cover form a GP:
There are infinitely many steps, and no step ever takes you to the wall. So it can look as though the room is impossible to cross.
But add up how far you have gone. After one step, . After two, . After three, . After four, . After ten, .
Those totals are creeping up on and never passing it. The pattern in the numerators is unmistakable: each total is , one short of the whole in a denominator that doubles every time.
So the sum of infinitely many shrinking distances is a perfectly finite number. The room is one room wide, and the infinitely many steps fit inside it with nothing left over.
The same arithmetic answers the bouncing ball. Its rebound heights , , , go on for ever in the model, and the total distance travelled is finite — the ball comes to rest in a finite time having covered a finite path, even though the model gives it infinitely many bounces. Nothing contradictory is happening; the bounces get shorter faster than they get more numerous.
And the repeating decimal from the numbers chapter is the same object once more. means , a GP with , and its total is exactly . Infinitely many terms, one clean answer — which is why the sum of a GP with gets a formula of its own in Class 11.
Exam relevance
How is the geometric progression tested in JEE Main?
Because the GP is the second standard progression, its sum formula is a Class 11 staple, and geometric growth is the model behind several chapters in other subjects.
This is the foundation for Class 10 and Class 11 Mathematics Sequences and Series, examined in JEE Main. The formula carries over unchanged, and Class 11 adds the sums:
The second one is the room-crossing puzzle in the previous section turned into a formula — with and it returns exactly . The Class 9 observation that the partial totals creep up on a limit without passing it is the whole content of that formula, and students who have seen meet it as a confirmation rather than a surprise.
The geometric mean pairs with the arithmetic mean. Class 11 defines the GM of and as — the middle term of a three-term GP — and the inequality is a recurring JEE Main tool for maxima and minima. That same appeared as the side of a square equal in area to a rectangle in the earlier area chapter, which is not a coincidence.
Repeating decimals become infinite GPs. The conversion done by the subtraction in the numbers chapter is the geometric sum formula in disguise, and Class 11 asks for it both ways.
Where geometric growth is used in other subjects. Class 11 Chemistry Nuclear Chemistry and Class 12 Physics Nuclei handle half-life, which is the halving GP of this page: a quantity reduced to after half-lives. These are examined in JEE Main and NEET, and the arithmetic is identical to worked example 5 above. Class 11 Chemistry Chemical Kinetics uses the same structure for first-order reactions.
Compound interest is the financial version, and the distinction drawn here — that three years of per cent loss gives , not — is the whole difference between simple and compound treatment.
What the questions look like. For board work, expect **test whether a sequence is a GP and state and , find a specified term, find which term equals a value, a growth or decay word problem, and a self-similar pattern needing the sequence of parts or of area. For JEE Main, expect sums of GPs, infinite sums, mixed AP-GP problems, and the AM-GM inequality.
How board and competitive emphasis differ. A board paper rewards the step-by-step values alongside the formula. A competitive paper assumes both formulas and tests whether you notice that an infinite sum converges, or that a word problem is geometric rather than arithmetic.
The single trap that costs the most marks.** Treating a repeated percentage change as a single larger one. Losing per cent a year for three years leaves per cent, not per cent, because each loss is taken from a smaller base. The defence is to write out the three years separately once — — which both gives the answer and shows why the shortcut fails.
This is the foundation for Class 10 and Class 11 Mathematics Sequences and Series, examined in JEE Main. The formula carries over unchanged, and Class 11 adds the sums:
The second one is the room-crossing puzzle in the previous section turned into a formula — with and it returns exactly . The Class 9 observation that the partial totals creep up on a limit without passing it is the whole content of that formula, and students who have seen meet it as a confirmation rather than a surprise.
The geometric mean pairs with the arithmetic mean. Class 11 defines the GM of and as — the middle term of a three-term GP — and the inequality is a recurring JEE Main tool for maxima and minima. That same appeared as the side of a square equal in area to a rectangle in the earlier area chapter, which is not a coincidence.
Repeating decimals become infinite GPs. The conversion done by the subtraction in the numbers chapter is the geometric sum formula in disguise, and Class 11 asks for it both ways.
Where geometric growth is used in other subjects. Class 11 Chemistry Nuclear Chemistry and Class 12 Physics Nuclei handle half-life, which is the halving GP of this page: a quantity reduced to after half-lives. These are examined in JEE Main and NEET, and the arithmetic is identical to worked example 5 above. Class 11 Chemistry Chemical Kinetics uses the same structure for first-order reactions.
Compound interest is the financial version, and the distinction drawn here — that three years of per cent loss gives , not — is the whole difference between simple and compound treatment.
What the questions look like. For board work, expect **test whether a sequence is a GP and state and , find a specified term, find which term equals a value, a growth or decay word problem, and a self-similar pattern needing the sequence of parts or of area. For JEE Main, expect sums of GPs, infinite sums, mixed AP-GP problems, and the AM-GM inequality.
How board and competitive emphasis differ. A board paper rewards the step-by-step values alongside the formula. A competitive paper assumes both formulas and tests whether you notice that an infinite sum converges, or that a word problem is geometric rather than arithmetic.
The single trap that costs the most marks.** Treating a repeated percentage change as a single larger one. Losing per cent a year for three years leaves per cent, not per cent, because each loss is taken from a smaller base. The defence is to write out the three years separately once — — which both gives the answer and shows why the shortcut fails.
Key takeaways
Geometric progressions, growth, decay and self-similarity: quick revision
- A sequence is a GP when consecutive ratios are all equal; that value is the common ratio .
- has ; has ; has ; has .
- **A negative alternates the signs** — odd terms positive, even terms negative.
- is neither an AP nor a GP. is an AP, not a GP.
- Two terms never settle it: and start identically.
- From two terms: second term , fifth term gives , so and .
- **Neither nor may be zero.
- th term**: — the fourth term needs three multiplications.
- of is ; of is ; of is .
- Locating a term: gives , so . is not a term.
- of is , since is positive.
- A GP outruns an AP completely: from , the twentieth terms are and .
- Doubling bacteria: cells after hours — note the exponent is , since *after hours* is the start.
- Bouncing ball: cm, checked as .
- Depreciation: losing per cent a year gives — a fall of per cent, not .
- Compound growth: at per cent for years gives , against simple.
- Halving: g after days is g.
- How many steps: needs , so hours ( then ).
- Sierpinski triangle: pieces so , with ; area , so after four stages; side .
- Sierpinski carpet: pieces , so at stage ; area .
- Read whether pieces or area is wanted — triangles against of the area.
- **A GP with approaches zero without reaching it**, which is where the model parts company with a real ball.
- Halving distances give totals — infinitely many terms, a finite total.
Fold a sheet of paper in half as many times as you can, count the layers at each fold, and work out how many layers the eighth fold would need.
- has ; has ; has ; has .
- **A negative alternates the signs** — odd terms positive, even terms negative.
- is neither an AP nor a GP. is an AP, not a GP.
- Two terms never settle it: and start identically.
- From two terms: second term , fifth term gives , so and .
- **Neither nor may be zero.
- th term**: — the fourth term needs three multiplications.
- of is ; of is ; of is .
- Locating a term: gives , so . is not a term.
- of is , since is positive.
- A GP outruns an AP completely: from , the twentieth terms are and .
- Doubling bacteria: cells after hours — note the exponent is , since *after hours* is the start.
- Bouncing ball: cm, checked as .
- Depreciation: losing per cent a year gives — a fall of per cent, not .
- Compound growth: at per cent for years gives , against simple.
- Halving: g after days is g.
- How many steps: needs , so hours ( then ).
- Sierpinski triangle: pieces so , with ; area , so after four stages; side .
- Sierpinski carpet: pieces , so at stage ; area .
- Read whether pieces or area is wanted — triangles against of the area.
- **A GP with approaches zero without reaching it**, which is where the model parts company with a real ball.
- Halving distances give totals — infinitely many terms, a finite total.
Fold a sheet of paper in half as many times as you can, count the layers at each fold, and work out how many layers the eighth fold would need.