Geometric sequences

Geometric sequences (GPs): the common ratio, testing for a GP, the formula T(n) = ar^(n−1), negative ratios and alternating signs, and finding which term reaches a given size.

Common ratio

A geometric sequence (or geometric progression, GP) multiplies by the same number each time. That number is the common ratio $r$.

The nth term

nth term of a GP

$$T_n = ar^{n-1}$$

As with an AP, there are $n - 1$ steps from $T_1$ to $T_n$, so $r$ is multiplied in $n - 1$ times.

Example 1●

Find $T_8$ for the GP $5, 10, 20, \ldots$

$a = 5$ and $r = 2$, so $T_8 = 5 \times 2^7 = 640$.

Example 2: a fractional ratio●●

Find $T_6$ for the GP $48, 24, 12, \ldots$

$r = \dfrac{24}{48} = \dfrac12$, so $T_6 = 48 \times \left(\dfrac12\right)^5 = \dfrac{48}{32} = \dfrac32$.

Negative ratios

When $r$ is negative, the signs alternate. For example, $3, -6, 12, -24, \ldots$ has $r = -2$. The terms with odd $n$ have the same sign as $a$; the terms with even $n$ have the opposite sign.

A GP with a positive ratio lies on an exponential curve

With r > 1 the terms grow, and with 0 < r < 1 they shrink towards 0. (A negative ratio makes the terms jump from side to side, so they do not lie on one curve.)

Try this

Set $r = 0.5$. What do the terms get close to?

Which term first passes a value?

Example 3●●●

Which is the first term of $2, 6, 18, \ldots$ to exceed $10\,000$?

$T_n = 2 \times 3^{n-1} > 10\,000$, so $3^{n-1} > 5000$.

$n - 1 > \dfrac{\ln 5000}{\ln 3} \approx 7.75$, so $n > 8.75$.

Take logs of both sides (or use trial and error on a calculator).

The first such term is $T_9 = 2 \times 3^8 = 13\,122$.

Flashcards

Common ratio

$r = \dfrac{T_{n+1}}{T_n}$

$n$th term of a GP

$T_n = ar^{n-1}$

$r < 0$

The signs alternate.

Practice

Find $T_8$ for $5, 10, 20, \ldots$
Find $T_6$ for $48, 24, 12, \ldots$
Which term of $2, 6, 18, \ldots$ is the first to exceed $10\,000$?
Which sequence is geometric?
Find $T_5$ for $3, -6, 12, \ldots$
In a GP with $a > 0$ and $r < 0$, which terms are negative?
A GP has $T_2 = 12$ and $T_4 = 108$, with positive terms. Find $r$.
A GP has $T_2 = 12$ and $T_4 = 108$, with positive terms. Find $a$.
The $n$th term of $81, 27, 9, \ldots$ is: