Which term of the GP: 2, 8, 32, ... is 131072? Write the explicit formula as well as the recursive formula Class 9

Which term of the GP: 2, 8, 32, ... is 131072? Write the explicit formula as well as the recursive formula Class 9

Question 1.

Which term of the GP: 2, 8, 32, ... is 131072? Write the explicit formula as well as the recursive formula for the n^th term. Class 9

Solution:

2, 8, 32,...

a = 2, r = $\frac{8}{2}$ = 4

∴ $t_n=2 \times 4^{n-1}$ is the explicit formula for n^th term.

Suppose n^th term is 131072,

∴ t_n = 131072

∴ 131072 = 2 × 4^n-1

∴ $\frac{131072}{2}$ = 4^n-1

∴ 65536 = 4^n-1

$\therefore \quad 4^8=4^{n-1} \quad \ldots\left[\left(\because 65536=2^{16}=\left(2^2\right)^8=4^8\right)\right]$

∴ n - 1 = 8

∴ n = 9

∴ 9^th term of GP is 131072.

t₁ = 2

t₂ = 4 × 2 = 8

t₃ = 4 × 8 = 32

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t_n = 4 × t_n-1

∴ The recursive formula for the n^th term is t₁ = 2, t_n = 4t_n-1 for n ≥ 2.