Why Adding a Half, a Quarter, an Eighth and So On Never Passes 1

A sequence is an ordered list of numbers — a first term, a second term, and so on. A series is what you get when you add those terms.

$$\text{sequence: }2,5,10,17,\dots \text{series: }2+5+10+17+\dots$$

This chapter treats sequences as functions, focuses on geometric progressions and their sums — including sums that go on forever — and compares the arithmetic and geometric means of two numbers.

**A sequence assigns a number $a_n$ to each natural number $n$, so it is a function with domain $\mathbb{N}$; its terms come either from an explicit rule for $a_n$ or from a recurrence that builds each term from earlier ones.

Worked example 1 — explicit rule.** $a_n=n^2+1$:

$$a_1=2,a_2=5,a_3=10,a_4=17$$

The series of the first four terms is $2+5+10+17=34$.

Worked example 2 — recurrence. $a_1=3$ and $a_n=3a_{n-1}+2$:

$$3,3(3)+2=11,3(11)+2=35,3(35)+2=107$$

Worked example 3 — two earlier terms. $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}$ gives $1,1,2,3,5,8,13,\dots$

Sigma notation. ${\sum }_{k=1}^4(k^2+1)=34$.

An everyday example. A diary of daily savings is a sequence; the total saved at the end of the month is the series.

The substance. A sequence depends on order, while a finite series has a single value whatever order its terms are added in.

**A sequence is a G.P. when each term divided by the previous one gives the same ratio $r$, and its $n$th term is $a_n=a{r}^{n-1}$.

Worked example 1.** $5,25,125,\dots$ has $r=5$, so

$$a_{10}=5\times 5^9=5^{10}=9765625$$

Worked example 2. Which term of $2,2\sqrt{2},4,\dots$ is $128$? Here $r=\sqrt{2}$.

$$2(\sqrt{2}{)}^{n-1}=128(\sqrt{2}{)}^{n-1}=64=(\sqrt{2}{)}^{12}n=13$$

Worked example 3. The $4$th term is $54$ and the $7$th is $1458$.

$$\frac{a{r}^6}{a{r}^3}=r^3=27r=3a=\frac{54}{27}=2$$

Worked example 4 — negative ratio. $3,-6,12,-24,\dots$ has $r=-2$.

An everyday example. **A culture of $30$ bacteria that doubles every hour** has $30\times 2^5=960$ bacteria after $5$ hours.

The substance. **Dividing two given terms removes $a$**, leaving a power of $r$ — the fastest way to find the ratio.

**For $r\ne 1$, $S_n=\frac{a(r^n-1)}{r-1}$; when $|r|<1$, the terms shrink towards zero and the infinite sum is $S_{\infty }=\frac{a}{1-r}$.

Worked example 1.** $1+3+9+\dots$ to $8$ terms:

$$S_8=\frac{3^8-1}{3-1}=\frac{6560}{2}=3280$$

Worked example 2 — infinite.

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots =\frac{\frac{1}{2}}{1-\frac{1}{2}}=16+1.2+0.24+\cdots =\frac{6}{1-0.2}=7.5$$

Worked example 3 — recurring decimals.

$$0.\overline{45}=0.45+0.0045+\cdots =\frac{0.45}{1-0.01}=\frac{45}{99}=\frac{5}{11}$$

Worked example 4 — a bouncing ball. A ball dropped from $10$ m rebounds to $\frac{3}{5}$ of its height each time. Total distance:

$$10+2\left(6+3.6+2.16+\dots \right)=10+2\times \frac{6}{1-0.6}=10+30=40\text{ m}$$

An everyday example. A rubber ball on a school playground travels a finite total distance even though, in this model, it bounces endlessly.

The boundary case. **If $|r|\ge 1$, there is no infinite sum**: $1+2+4+\dots$ grows without limit.

**For positive numbers $a$ and $b$, $\text{A.M.}=\frac{a+b}{2}$ and $\text{G.M.}=\sqrt{ab}$; inserting $k$ geometric means makes a G.P. of $k+2$ terms; and $\text{A.M.}\ge \text{G.M.}$, with equality only when $a=b$.

Worked example 1.** For $4$ and $16$: $\text{A.M.}=10$, $\text{G.M.}=8$, and $10\ge 8$.

Worked example 2 — inserting means. Insert $3$ geometric means between $1$ and $256$. The G.P. has $5$ terms:

$$256=1\times r^{4}r=4\text{means: }4,16,64$$

**Why A.M. $\ge$ G.M.** For positive $a,b$:

$$(\sqrt{a}-\sqrt{b}{)}^2\ge 0\Rightarrow a+b\ge 2\sqrt{ab}$$

Worked example 3 — a minimum. For $x>0$, $x+\frac{1}{x}\ge 2\sqrt{x\cdot \frac{1}{x}}=2$, with the minimum $2$ at $x=1$.

Worked example 4 — from the means. If $\text{A.M.}=10$ and $\text{G.M.}=8$, then $a+b=20$ and $ab=64$, so the numbers are $16$ and $4$.

An everyday example. **A rectangular kitchen garden of area $36{\text{ m}}^2$** needs fencing $2(l+w)\ge 2\times 2\sqrt{36}=24$ m, and the least fencing is used when it is a $6$ m square.

The substance. The G.M. is defined here only for positive numbers, since $\sqrt{ab}$ must be real.