Mathematics · Ch 8 — Sequences and Series

Geometric Progression (G.P.)

8.4

Geometric Progression (G.P.)

Consider these three sequences:

$2, 4, 8, 16, \dots$
$1, -\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \frac{1}{81}, \dots$
$0.1, 0.01, 0.001, 0.0001, \dots$ What pattern governs how the terms progress? In each sequence, every term after the first is obtained by multiplying the previous term by a fixed number. In (i), each term is $2$ times the one before it. In (ii), each term is $-\frac{1}{3}$ times the previous term. In (iii), each term is $0.01$ times the previous term. This constant multiplier is called the common ratio, and such sequences are called geometric progressions (abbreviated as G.P.).
Note

A geometric progression is defined only when every term is non-zero. If any term were zero, the ratio with the preceding term would be undefined.

Formal Definition

A sequence $a_1, a_2, a_3, \dots, a_n, \dots$ is called a geometric progression if each term is non-zero and

\frac{a_{k+1}}{a_k} = r \quad (\text{constant}), \text{ for } k \geq 1

The constant $r$ is called the common ratio of the G.P.

If we denote the first term by $a$ (so $a_1 = a$ ), then the terms of the G.P. are:

a, \; ar, \; ar^2, \; ar^3, \; ar^4, \dots

Here:

$a$ = first term
$r$ = common ratio

For the three examples above:

Sequence (i): $a = 2$ , $r = 2$
Sequence (ii): $a = 1$ , $r = -\frac{1}{3}$
Sequence (iii): $a = 0.1$ , $r = 0.01$

Watch out

The common ratio can be negative, as in example (ii). This causes the terms to alternate in sign. A negative common ratio does not make the sequence invalid — it simply produces an alternating G.P.

Notation Used in Formulae

When working with geometric progressions, we use the following standard notation:

Symbol	Meaning
$a$	first term
$r$	common ratio
$l$	last term (when the number of terms is finite)
$n$	number of terms
$S_n$	sum of the first $n$ terms

Important

The $n$ th term of a G.P. is $ar^{n-1}$ , not $ar^n$ . This is a common source of error. The first term corresponds to $n=1$ , giving $ar^{0} = a$ .

The $n$ th Term of a G.P.

From the pattern $a, ar, ar^2, ar^3, \dots$ , we see that:

$a_1 = a = ar^{0}$
$a_2 = ar = ar^{1}$
$a_3 = ar^2$
$a_4 = ar^3$

In general, the $n$ th term (also called the general term) is:

a_n = ar^{n-1}

If the G.P. has a finite number of terms and the last term is denoted by $l$ , then:

l = ar^{n-1}

Tip

To find the $n$ th term quickly, identify $a$ and $r$ first, then substitute into $ar^{n-1}$ . For example, in the G.P. $3, 6, 12, 24, \dots$ , we have $a=3$ and $r=2$ , so the 10th term is $3 \cdot 2^{9} = 3 \cdot 512 = 1536$ .

Sum of $n$ Terms of a G.P.

We now derive a formula for the sum of the first $n$ terms of a G.P.

Let $S_n$ denote the sum of the first $n$ terms:

S_n = a + ar + ar^2 + \dots + ar^{n-2} + ar^{n-1}

Multiply both sides by $r$ :

rS_n = ar + ar^2 + ar^3 + \dots + ar^{n-1} + ar^{n}

Subtract $rS_n$ from $S_n$ :

S_n - rS_n = a - ar^{n}

S_n(1 - r) = a(1 - r^{n})

Therefore, when $r \neq 1$ :

S_n = \frac{a(1 - r^{n})}{1 - r}

If $r = 1$ , then every term is $a$ , and the sum is simply:

S_n = a + a + a + \dots + a = na

S_n = \begin{cases} \frac{a(1 - r^{n})}{1 - r}, & r \neq 1 \\ na, & r = 1 \end{cases}

Note

The formula $\frac{a(1 - r^{n})}{1 - r}$ is valid for any $r \neq 1$ , including negative values of $r$ . When $r > 1$ , it is often more convenient to write the formula as $\frac{a(r^{n} - 1)}{r - 1}$ to avoid a negative denominator.

Sum When the Last Term is Known

If we know the last term $l = ar^{n-1}$ , we can express the sum in an alternative form. Since $l = ar^{n-1}$ , we have $lr = ar^{n}$ . Substituting into the sum formula:

S_n = \frac{a - lr}{1 - r}, \quad r \neq 1

This form is useful when the last term is given directly rather than the number of terms. …

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Geometric Progression (G.P.)