A sequence is given by the recursive rule t1 = 2, tn+1 = 3tn-2 for n ≥ 1. Which term of the sequence Class 9

A sequence is given by the recursive rule t1 = 2, tn+1 = 3tn-2 for n ≥ 1. Which term of the sequence Class 9

Question 1.

A sequence is given by the recursive rule t₁ = 2, t_n+1 = 3t_n-2 for n ≥ 1. Which term of the sequence is 730? Class 9

Solution:

Here, t₁ = 2, t_n+1 = 3_n-2 for n ≥ 1

Finding the few terms

t₂ = 3(2) -2 = 4

t₃ = 3(4) - 2 = 10

t₄ = 3(10) - 2 = 28

t₅ = 3(28) - 2 = 82

t₆ = 3(82) - 2 = 244

t₇ = 3(244) - 2 = 730

∴ 730 is the 7^th term of the sequence.

Question 2.

Which term of the GP: 2, 6, 18,... is 4374? Write the explicit formula as well as the recursive formula for the nth term. Class 9

Solution:

Here, a = 2, r = $\frac{6}{2}$ = 3

t_n = ar^n-1

∴ t_n = 2 × 3^n-1

∴ t_n = 2 × 3^n-1 is the explicit formula for the n^n-1 term.

Suppose n^th term is 4374

∴ t_n = 4374

∴ 2 × 3^n-1 = 4374

∴ 3^n-1 = $\frac{4374}{2}$ = 2187

∴ 3^n-1 = 3⁷

∴ n - 1 = 7

∴ n = 8

∴ 4374 is the 8^th term of the GP.

Recursive formula is

t₁ = 2, t_n = 3 × t_n-1 for n ≥ 2