Question

The 3rd term of a geometric sequence is -2 and the 7th is -32. Find the common ratio,the first term, the explicit formula, and the 10th term.

Answer 1

see explanation

Explanation:

The n th term of a geometric sequence is

`a_(n)` = a
`(r)^(n-1)`

where a is the first term and r the common ratio

Both a and r have to be found

Given a₃ = - 2, then

ar² = - 2 → (1)

Given a₇ = - 32, then

a
`r^(6)` = - 32 → (2)

Divide (2) by (1)

`(ar^6)/(ar^2)` =
`(-32)/(-2)`, that is

`r^(4)` = 16 ( take the fourth root of both sides )

r = 2 ← common ratio

Substitute r = 2 into (1)

a × 2² = - 2, that is

4a = - 2 ( divide both sides by 4 )

a = -
`(1)/(2)` ← first term

Hence

`a_(n)` = -
`(1)/(2)`
`(2)^(n-1)` ← explicit formula

and

`a_(10)` = -
`(1)/(2)` ×
`2^(9)` = - 0.5 × 512 = - 256