Question

Given the first term and the common ratio of a geometric sequence find the explicit formula.

Answer 1

A geometric sequence is generated by multiplying the first term in the sequence, $a_1$, several times by a fixed scalar $r$. So

$a_2=a_1r$

$a_3=a_2r=a_1r^2$

$a_4=a_3r=a_1r^3$

and so on, down to

$a_n=a_{n-1}r=a_{n-2}r^2=\cdots=a_1r^{n-1}$

Using the result above, the sequences you're interested in are

31) $a_n=-(-2)^{n-1}$

33) $a_n=4(-4)^{n-1}$

We can rewrite these results to make them slightly cleaner.

$-(-2)^{n-1}=-(-1)^{n-1}2^{n-1}=(-1)^n2^{n-1}$

$4(-4)^{n-1}=4(-1)^{n-1}4^{n-1}=(-1)^{n-1}4^n$