Question

What is the recursive formula for this geometric sequence? -3, -21, -147, -1029, ...

Answer 1

$a_{n+1} =7a_{n}$.

If we know the term $n^{th}$ and the common relation, r, of a geometric sequence, you can find the term $(n+1)^{th}$ using the recursive formula $a_{n+1} =a_{n}.r$.

The first term of the geometric sequence is a₁ = -3.

The common relation we have to find the relationship between a term and the term that precedes it.

$r=\frac{-21}{-3} = 7$

The recursive formula is:

$a_{1} =-3$

$a_{n+1} =7a_{n}$

Answer 2

$a_1=-3\\r=7\\a_n=a_{n-1}\cdot r\\\\ \boxed{a_n=7a_{n-1}}$