Question

Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. (Assume that n begins with 1.) $-5,\frac{10}{3},\frac{-20}{9},\frac{40}{27},\frac{-80}{81},...$

Answer 1

The formula for the general term of the sequence :

$T_n=-5{\left(\frac{-2}{3}\right)}^{ⁿ}wheren=0,1,2,3.......$

Given sequence is: $-5,\frac{10}{3},\frac{-20}{9},\frac{40}{27},\frac{-80}{81},...$

The ratio of second to first term$=\frac{-2}{3}$

The ratio of second to the third term $=\frac{-2}{3}$

The ratio of the third to the fourth term$=\frac{-2}{3}$

Looking at the pattern, we can say that the given sequence is a Geometric Progression with the first term as $-5$ and the common ratio is $\frac{-2}{3}$

What is the nth term of a GP?

The nth term of a GP with first term a and common ratio r is $arⁿ.$

So, the formula for the nth term for the given sequence will be:

$T_n=-5{\left(\frac{-2}{3}\right)}^{ⁿ}wheren=0,1,2,3.......$

Therefore, the formula for the general term of the sequence is

$T_n=-5{\left(\frac{-2}{3}\right)}^{ⁿ}wheren=0,1,2,3.......$