Question

Given these two terms in a geometric sequence, find the recursive formula.
a1=4
a6=31104

Answer 1

`\boxed{ a_(n) = 6a_(n-1)}`

Explanation:

Step 1. Determine the common ratio

The formula for the nth term of a geometric sequence is

aₙ = a₁rⁿ⁻¹

Data:

a₁ = 4

n = 6

a₆ =31 104

Calculation:

31 104 = 4r⁵

r⁵ = 7776

`r = \sqrt [5]{7776}`

r = 6

aₙ = 4(6)ⁿ

Step 2. Determine the recursive formula.

aₙ = 4(6)ⁿ

aₙ₋₁ = 4(6)ⁿ⁻¹

`\frac{a_(n)}{{a_(n-1)}} = (4(6)^(n) )/(4(6)^(n-1)) = 6\\\\a_(n) = 6a_(n-1)`

The recursive formula for the series is
`\boxed{ a_(n) = 6a_(n-1)}`