4.4.1 Geometric Sequences

Geometric Sequences

A geometric sequence is one that has a common ratio or multiplier between terms.

e.g. $2, 4, 8, 16, 32, \ldots$

(a = 2, \quad r = 2) \quad \text{where } r \text{ is the common ratio between terms}

e.g.\ 100, 50, 25, 12.5, \ldots

(a = 100, \quad r = \frac{1}{2})

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We can obtain the common ratio between terms by dividing a term by its previous term.

If ' $a$ ' denotes the first term and ' $r$ ' the common ratio between terms, then:

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\begin{align*} u_1 &= a \\ u_2 &= ar \\ u_3 &= ar^2 \\ &\vdots \\ u_n &= ar^{n-1} \end{align*}

For a geometric sequence:

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u_n = ar^{n-1} \quad \text{(Must remember this)}

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S_n = \frac{a(1 - r^n)}{1 - r} \quad \text{or} \quad \frac{a(r^n - 1)}{r - 1} \quad \text{(Given in formulae)}

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\begin{align*} S_n &= a + ar + ar^2 + \ldots + ar^{n-1} \\ rS_n &= ar + ar^2 + \ldots + ar^{n-1}+ar^n \\ S_n - rS_n &= a + 0 + 0 + \ldots + 0 - ar^n \\ \Rightarrow S_n - rS_n &= a - ar^n \\ \Rightarrow S_n (1 - r) &= a (1 - r^n) \\ \Rightarrow S_n &= \frac{a (1 - r^n)}{1 - r} \quad \text{as required} \end{align*}

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For each of the following geometric progressions, find an expression for the $n^{th}$ term. a) $1, 5, 25, 125, \ldots$

a = 1, \quad r = 5

\Rightarrow u_n = 1 \times 5^{n-1} = 5^{n-1}

b) $3, -12, 48, -192, \ldots$

a = 3, \quad r = -4

u_n = 3 \times (-4)^{n-1}

\sum_{r=1}^{6} 8^{r+1} = 64 + 512 + \ldots

a = 64, \quad r = 8

S_6 = \frac{64(1-8^6)}{1-8}

= 2396736 \quad \text{(Check)}

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The second and fifth terms of a geometric series are 0.5 and 32 respectively. a. Find the first term and common ratio of the series.

b. Find the number of terms of the series that are smaller than 10,000.

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Solution:

ar = 0.5 \quad \text{(A)}

ar^4 = 32 \quad \text{(B)}

To find the ratio, divide (B) by (A) because (B) has the higher power of $r$ :

\frac{\cancel ar^4}{\cancel ar} = \frac{32}{0.5}\quad B \div A

\Rightarrow r^3 = 64 \Rightarrow r = \sqrt[3]{64} = 4

Using $ar = 0.5$ :

a(4) = 0.5 \Rightarrow a = \frac{1}{8}

b) Finding the number of terms:

ar^{n-1} < 10,000

\Rightarrow \frac{1}{8}(4)^{n-1} < 10,000

4^{n-1} < 80,000 \quad \text{(Requires logarithms)}

\Rightarrow \log_4(4^{n-1}) < \log_4(80,000)

\Rightarrow n-1 < \log_4(80,000)

\Rightarrow n < \log_4(80,000) + 1

\Rightarrow n < 9.14

\Rightarrow n = 9 \quad \text{(n must be an integer)}