a) If $3^{2n+2} - 8n - 9$ is divisible by $k$ for all $n \in N$ is true, then which one of the following is a value of $k$? (1)
i) $8$ ii) $6$ iii) $3$ iv) $12$
b) Prove by using the principle of Mathematical Induction $P(n) = 1 + 3 + 3^2 + \ldots + 3^{n-1} = \dfrac{3^n - 1

Question

a) If $3^{2n+2} - 8n - 9$ is divisible by $k$ for all $n \in N$ is true, then which one of the following is a value of $k$? (1)
i) $8$   ii) $6$   iii) $3$   iv) $12$
b) Prove by using the principle of Mathematical Induction $P(n) = 1 + 3 + 3^2 + \ldots + 3^{n-1} = \dfrac{3^n - 1}{2}$ is true for all $n \in N$. (3)

Accepted Answer

>

> [!TLDR]
> Testing small n shows 3^(2n+2) − 8n − 9 is a multiple of 8; the sum identity is proved by the standard induction steps (base + inductive).
>
> a) Put n = 1: 3⁴ − 8 − 9 = 81 − 17 = 64. Put n = 2: 3⁶ − 16 − 9 = 729 − 25 = 704. Both are divisible by 8 (64 = 8·8, 704 = 8·88), while 6, 3, 12 do not divide 64. So k = 8 → option (i).
>
> b) Let P(n): 1 + 3 + 3² + … + 3^(n−1) = (3ⁿ − 1)/2.
>
> Base step (n = 1): LHS = 1; RHS = (3¹ − 1)/2 = 2/2 = 1. True.
>
> Inductive step: assume P(k) true, i.e. 1 + 3 + … + 3^(k−1) = (3ᵏ − 1)/2.
> Add the next term 3ᵏ to both sides: …

Q.a) If $3^{2n+2} - 8n - 9$ is divisible by $k$ for all $n \in N$ is true, then which one of the following is a value of $k$ ?

Concept understanding — Proving Divisibility by Induction

Q.a) If 32n+2−8n−93^{2n+2} - 8n - 932n+2−8n−9 is divisible by kkk for all n∈Nn \in Nn∈N is true, then which one of the following is a value of kkk?

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Q.a) If $3^{2n+2} - 8n - 9$ is divisible by $k$ for all $n \in N$ is true, then which one of the following is a value of $k$ ?

Unlock everything free for 14 days