1.4 Geometric Sequence (G.S.)

1. The Definition

If the first term of a sequence is $u_1 = 5$, and it is multiplied by a fixed number, say $r=2$, to find the next term, the following sequence is generated: $5, 10, 20, 40, 80, \dots$. Such a sequence is called geometric. In a geometric sequence, the ratio between any two consecutive terms is constant.

Only two elements are needed: The first term $u_1$, and the common ratio $r$.

EXAMPLE 1

  • (a) If $u_1 = 1, r=2$, the sequence is $\mathbf{1, 2, 4, 8, 16, 32, 64, \dots}$
  • (b) If $u_1 = 5, r=10$, the sequence is $\mathbf{5, 50, 500, 5000, \dots}$
  • (c) If $u_1 = 1, r=-2$, the sequence is $\mathbf{1, -2, 4, -8, 16, \dots}$
  • (d) If $u_1 = 1, r=\frac{1}{2}$, the sequence is $\mathbf{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots}$
  • (e) If $u_1 = 1, r=-\frac{1}{2}$, the sequence is $\mathbf{1, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \frac{1}{16}, -\frac{1}{32}, \dots}$

NOTICE:

  • The common ratio $r$ may also be negative. In this case the signs alternate $(+, -, +, -, \dots)$.
  • The common ratio $r$ may be between -1 and 1, that is $|r| < 1$. In such a sequence the terms approach 0.

2. QUESTION A: What is the general formula for $u_n$?

If $u_1$ and $r$ are known, then the general formula is:

$u_n = u_1 r^{n-1}$

The reasoning is as follows: In order to find $u_5$, one starts from $u_1$ and then multiplies 4 times by the ratio $r$.

Hence, $u_5 = u_1 r^4$
Similarly, $u_{10} = u_1 r^9$
$u_{50} = u_1 r^{49}$
In general, $u_n = u_1 r^{n-1}$

EXAMPLE 2

In a geometric sequence let $u_1 = 3$ and $r = 2$. Find (a) the first four terms and (b) the 100th term.

Solution:

(a) $\mathbf{3, 6, 12, 24}$
(b) The general formula is needed: $u_{100} = u_1 r^{99} = \mathbf{3 \cdot 2^{99}}$. (This answer is sufficient as it is very large).

EXAMPLE 3

In a geometric sequence let $u_1 = 10$ and $u_{10} = 196830$. Find $u_3$.

Solution:

Since $u_1$ is known, $r$ is needed. The information for $u_{10}$ is exploited first.
$u_{10} = u_1 r^9 \quad \Rightarrow \quad 196830 = 10 \cdot r^9$
$r^9 = 19683$
$r = \sqrt[9]{19683} = 3$
Therefore, $u_3 = u_1 r^2 = 10 \cdot 3^2 = \mathbf{90}$

⚠️ REMEMBER

The first task in a G.S. is to find the basic elements, $u_1$ and $r$, and then everything else!

EXAMPLE 4

A geometric sequence has a fifth term of 3 and a seventh term of 0.75. Find (a) the first term $u_1$ and the common ratio $r$, and (b) the tenth term.

Solution:

(a) It is given that $u_5 = 3$ and $u_7 = 0.75$. The formula takes the form:
$u_5 = u_1 r^4$
$u_7 = u_1 r^6$
Divide $u_7$ by $u_5$:
$\frac{u_7}{u_5} = \frac{u_1 r^6}{u_1 r^4}$
$\frac{0.75}{3} = r^2$
$r^2 = 0.25 \quad \Rightarrow \quad r = \mathbf{\pm 0.5}$
The first equation then gives:
$3 = u_1 (0.0625) \quad \Rightarrow \quad u_1 = \frac{3}{0.0625} = \mathbf{4.8}$
(b) The tenth term is $u_{10} = u_1 r^9$:
If $r = 0.5$, then $u_{10} = 4.8(0.5)^9 = \mathbf{0.09375}$
If $r = -0.5$, then $u_{10} = 4.8(-0.5)^9 = \mathbf{-0.09375}$
Notice: Two sequences satisfy the criteria:
$1^{st}: 4.8, 24, 12, 6, 3, 1.5, 0.75 \dots (r = 0.5)$
$2^{nd}: 4.8, -24, 12, -6, 3, -1.5, 0.75 \dots (r = -0.5)$

3. QUESTION B: What is the sum $S_n$ of the first $n$ terms?

Given that $r \neq 1$, the result is given by:

$S_n = \frac{u_1(r^n - 1)}{r - 1}$
or $\quad S_n = \frac{u_1(1 - r^n)}{1 - r}$

Proof for $S_n$ (mainly for Math HL)

By definition: $S_n = u_1 + u_1 r + u_1 r^2 + \dots + u_1 r^{n-2} + u_1 r^{n-1} \quad \text{(1)}$
Multiply by $r$: $r \cdot S_n = u_1 r + u_1 r^2 + u_1 r^3 + \dots + u_1 r^{n-1} + u_1 r^n \quad \text{(2)}$
Subtracting (2) - (1) gives:
$r \cdot S_n - S_n = u_1 r^n - u_1$
$(r - 1)S_n = u_1(r^n - 1)$
$S_n = \frac{u_1(r^n - 1)}{r - 1}$

EXAMPLE 5

Consider the sum $2 + 2^2 + 2^3 + \dots + 2^{10}$.

Solution:

It is a geometric series of 10 terms with $u_1 = 2$ and $r = 2$. The sum is:
$S_{10} = \frac{u_1(r^{10} - 1)}{r - 1} = \frac{2(2^{10} - 1)}{2 - 1} = \mathbf{2046}$

EXAMPLE 6

Consider the sum $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^{10}}$.

Solution:

It is a geometric series of 11 terms with $u_1 = 1$ and $r = 1/2$. The sum is given by (preferring the second version of $S_n$):
$S_{11} = \frac{u_1(1 - r^{11})}{1 - r} = \frac{1 \cdot (1 - (\frac{1}{2})^{11})}{1 - \frac{1}{2}} = \frac{1 - \frac{1}{2048}}{\frac{1}{2}} = \mathbf{\frac{2047}{1024}}$

EXAMPLE 7

In a G.S. $u_2 = -30$ and $S_2 = -15$. Find $u_1$ and $r$.

Solution:

$S_2 = u_1 + u_2 \quad \Rightarrow \quad -15 = u_1 - 30 \quad \Rightarrow \quad u_1 = \mathbf{15}$
Since $u_1 = 15$ and $u_2 = -30$:
$r = \frac{u_2}{u_1} = \frac{-30}{15} = \mathbf{-2}$
[Indeed, it is the G.S. $15, -30, 60, -120, \dots$ with $u_2 = -30, S_2 = -15$]

4. Notice for Consecutive Terms

Let $a, x, b$ be consecutive terms of a geometric sequence (regardless of whether these are the first three terms or some other three consecutive terms). The common ratio is equal to:

$\frac{x}{a} = \frac{b}{x}$
For example, if $10, x, 90$ are consecutive terms in a G.S.:
$\frac{x}{10} = \frac{90}{x} \quad \Rightarrow \quad x^2 = 900 \quad \Rightarrow \quad x = \mathbf{\pm 30}$ (two solutions)

5. The Sum of Terms in an Infinite G.S.

Consider the sum of the infinite geometric sequence $S_\infty = u_1 + u_2 + u_3 + \dots$ (it never stops!).

The result exists only if $-1 < r < 1$. It is given by the formula:

$S_\infty = \frac{u_1}{1 - r}$

In this case, the series is said to converge. Otherwise (if $|r| \ge 1$), the series diverges.

Three Proofs of the Formula (mainly for Math HL)

(a) Consider the formula for $S_n = \frac{u_1(r^n - 1)}{r - 1}$.
If $n \rightarrow \infty$ (i.e. $n$ tends to infinity), then $r^n \rightarrow 0$ (since $-1 < r < 1$).
Thus $S_n \rightarrow \frac{u_1(0 - 1)}{r - 1} = \frac{u_1}{1 - r}$.
(b) An alternative proof is similar to that for $S_n$:
$S_\infty = u_1 + u_1 r + u_1 r^2 + u_1 r^3 + \dots \quad \text{(1)}$
$r \cdot S_\infty = u_1 r + u_1 r^2 + u_1 r^3 + \dots \quad \text{(2)}$
Assuming that $S_\infty$ exists, subtract (1) - (2):
$S_\infty - r \cdot S_\infty = u_1 \quad \Rightarrow \quad (1 - r)S_\infty = u_1 \quad \Rightarrow \quad S_\infty = \frac{u_1}{1 - r}$.
(c) A slight modification of proof (b):
$S_\infty = u_1 + u_1 r + u_1 r^2 + u_1 r^3 + \dots$
$= u_1 + r(u_1 + u_1 r + u_1 r^2 + u_1 r^3 + \dots)$
$= u_1 + r S_\infty$
Hence, (assuming again that $S_\infty$ exists), $S_\infty - r S_\infty = u_1 \quad \Rightarrow \quad S_\infty = \frac{u_1}{1 - r}$.

EXAMPLE 8

Show that $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots = 1$.

Solution:

This is an infinite G.S. with $u_1 = \frac{1}{2}$ and $r = \frac{1}{2}$.
Since $|r| < 1$, the sum is obtained:
$S_\infty = \frac{u_1}{1 - r} = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = \mathbf{1}$.

EXAMPLE 9

Show that $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots = 2$.

Solution:

Method A: Just add 1 on both sides of the equation from the previous example.
Method B: It is an infinite G.S. with $u_1 = 1$ and $r = 1/2$. Hence,
$S_\infty = \frac{u_1}{1 - r} = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = \mathbf{2}$.

EXAMPLE 10

Show that $0.3333\dots = \frac{1}{3}$.

Solution:

It can be written as $0.3333\dots = 0.3 + 0.03 + 0.003 + \dots$.
This is in fact an infinite G.S. with $u_1 = 0.3$ and $r = 0.1$. Hence,
$0.3333\dots = \frac{u_1}{1 - r} = \frac{0.3}{1 - 0.1} = \frac{0.3}{0.9} = \mathbf{\frac{1}{3}}$.

EXAMPLE 11

Show that $0.9999\dots = 1$.

Solution:

Indeed, using the geometric series formula:
$0.9999\dots = 0.9 + 0.09 + 0.009 + \dots = \frac{u_1}{1 - r} = \frac{0.9}{1 - 0.1} = \frac{0.9}{0.9} = \mathbf{1}$.
Alternative Proof 1:
It is known that $0.3333\dots = \frac{1}{3}$. Multiplying both sides by 3 yields $0.9999\dots = \frac{3}{3} = \mathbf{1}$.
Alternative Proof 2:
Let $x = 0.9999\dots \quad \text{(1)}$
Then $10x = 9.9999\dots \quad \text{(2)}$
Subtracting (2) - (1) yields:
$10x - x = 9 \quad \Rightarrow \quad 9x = 9 \quad \Rightarrow \quad \mathbf{x = 1}$.
Thus, (1) and (3) give $0.9999\dots = 1$.