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Prove the following by using the principle of mathematical induction for all
is divisible by 8.
Let is divisible by 8.
I. For n = 1,
P(1) : is divisible by 8
is divisible by 8 81 - 17 is divisible by 8 64 is divisible by 8
which is true
∴ P(n) is true for n = 1
II. Let the statement be true for n = m,
∴ is divisible by 8
...(i)
III. For n = m + 1,
is divisible by 8
Now,
= [By (i)]
= 72k + 72m + 81 - 8m - 17 = 72k + 64m + 64 = 8(9k + 8m + 8),
= 8k' where k' = 9k + 8m +
∴ is divisible by 8.
P(m + 1) is true.
∴ P(m) is true P (m + 1) is true.
Hence, by the principle of mathematical induction, P(n) is true for all
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