Using the Mathematical Induction to prove that: 3^(2n)-1 is divisible by 4, whenever n is a positive integer.

Question

Using the Mathematical Induction to prove that:

3^{2 n} - 1

is divisible by 4, whenever n is a positive integer.

Answer 1

Claim: $3^{2 n} - 1$ is divisible by 4 for $n \in N$ .
We will prove this by induction.
Base step: $n = 1$
$∴ 3^{2} - 1 = 8$ , which is divisible by 4
$∴$ the result is true fir $n = 1.$
Inductive step:
Assure that result is true for $n = k$
i.e. $\frac{4}{3^{2 k}} - 1$ ...(1)
Now, we prove that: result is true form $= k + 1$
i.e. $\frac{4}{3^{2 (k + 1)}} - 1$
Consider $3^{2 (k + 1)} \equiv 3^{2 k} \cdot 9 (mod 4)$
$\equiv 1 \cdot 9 (\mod 4) [o m (i)]$
$∴ 3^{2 (k + 1)} \equiv 1 (mod 4)$
$∴ \frac{4}{3^{2 (k + 1)}} - 1$
$∴$ The result is true for $n = k + 1$ .
$∴$ By induction,
$3^{2 (n + 1)} - 1$ is divisibly by 4 for $n \in N$

Using the Mathematical Induction to prove that: 3^(2n)-1 is divisible by 4, whenever n is a positive integer.

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