If a_n = 24, which recursive formula could represent the sequence below? ...,24, 88, 664, 8408
A_n = (A_{n - 2})² + A_{n - 1}; A₁ = 2
A_n = 3A_{n - 1} + 16; A₁ = 16
A_n = (n)A_{n - 1} - 8; A₁ = 4
A_n = 2A_{n - 2} + 7A_n; A₁ = 2

Solution:

Given, the series is 24, 88, 664, 8408,...

a_n = 24

We have to find the recursive formula.

From the series, when n = 4

a_{n - 3} = a₁ = 24

a_{n - 2} = a₂ = 88

a_{n - 1} = a₃ = 664

a_n = a₄ = 8408

From option,

a) a_n = (a_{n - 2})² + a_{n - 1}; a₁ = 2

a_n = (88)² + 664

a_n = 7744 + 664

a_n = 8408

Therefore, option (a) is true.

b) a_n = 3a_{n - 1} + 16; a₁ = 16

a_n = 3(664) + 16

a_n = 1992 + 16

a_n = 2008

Therefore, option (b) is false.

c) a_n = (n)a_{n - 1} - 8

a₄ = (4)(664) - 8

a₄ = 2656 - 8

a₄ = 2648

Therefore, option (c) is false.

d) a_n = 2a_{n - 2} + 7a_{n - 1}

a_n = 2(88) + 7(664)

a_n = 176 + 4648

a_n = 4824

Therefore, option (d) is false.

Therefore, the recursive formula a_n = (a_{n - 2})² + a_{n - 1} represents the given sequence.

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If a_n = 24, which recursive formula could represent the sequence below? ...,24, 88, 664, 8408

Summary:

If a_n = 24, the recursive formula that represent the sequence 24, 88, 664, 8408,... is a_n = (a_{n - 2})² + a_{n - 1}.