Prove that for each positive integer n, the integer 32n+1 + 2n+2 is divisible by 7.
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To prove:
For every positive integer n, the integer will be divisible by 7
Proof:
Considering the method of mathematical induction
Let us check for n = 1
→ = = 35 (divisible by 7)
Checking for n = 2
→ = = 243 + 16 = 259 (divisible by 7)
Now, let us assume that
is divisible by 7 for some positive integer 'x' such that,
for any integer k.
that is,
Now, We need to show that integer will be divisible by 7 for n = x + 1 also, i.e., is also divisible by 7
so,
using equation (1)
That is the multiple of 7.
Therefore, By the concept of mathematical induction, is divisible by 7 for all positive integer n.
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