Use proof by mathematical induction to prove explicit formula for recursive sequence is correct

Question

Here is the question I am having trouble with.

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Suppose you have the following recursively defined sequence:

P₁ = 4
P_k = P_k-1+ 4 · 3^k for all integers k ≥ 2

Suppose you have used the method of iteration to find the following explicit formula:

P₁ = 4
P_n = 2 · 3ⁿ⁺¹ - 14 for all integers n ≥ 1

Use proof by mathematical induction to show this is the correct explicit formula.

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Here is what I have so far:

Base case: n = 1, 4 = 2 · 3¹⁺¹ - 14; 4 = 2 · 9 - 14; 4 = 18 - 14; 4 = 4 ✓

Assume: P_k = 2 · 3^k+1 - 14 for all integers k >= 1

Now we must show that P_k+1 = 2 · 3^(k+1)+1 - 14

Then we must get P_k+1 = P_k + 4 · 3^k to look like what was previously stated. First, let’s substitute in our assumption:

(2 · 3^k+1 - 14) + 4 · 3^k

I will say that’s as far as I got because I have tried simplifying this multiple ways and I cannot even get it close to what it should look like. I’m starting to think I’ve made a mistake elsewhere. Any help is appreciated, thanks!

Answer 1

Try using J.G.'s hint and then you can check your answer with the one below.

$P_k = 2 · 3^{k+1} - 14$

Therefore

$P_{k+1} = (2·3^{k+1} - 14)+4.3^{k+1}$

Add the $2·3^{k+1}$ and $4·3^{k+1}$ to get $6·3^{k+1}$

Then

$P_{k+1} = 6·3^{k+1} - 14=2·3^{k+2} - 14$

and you have the correct result.

Answer 2

You should have added $4\cdot 3^{k+1}$. It'll work now.