Question

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

Answer 1

Answer:

$\boxed{ a_{n} = 6a_{n-1}}$

Step-by-step 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

$\dfrac{a_{n}}{{a_{n-1}}} = \dfrac{4(6)^{n} }{4(6)^{n-1}} = 6\\\\a_{n} = 6a_{n-1}$

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