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प्रश्न
Answer the following:
Prove by method of induction 52n − 22n is divisible by 3, for all n ∈ N
उत्तर
Let P(n) ≡ 52n – 22n is divisible by 3, for all n ∈ N.
Step 1:
For n = 1, 52n – 22n = 52 – 22 = 25 – 4 = 21, which is divisible by 3.
∴ P(1) is true.
Step 2:
Let us assume that for some k ∈ N, P(k) is true, i.e. 52k – 22k is divisible by 3.
∴ `(5^(2"k") - 2^(2"k"))/3` = m (Say), whre m ∈ N
∴ 52k – 22k = 3m
∴ 52k = 22k + 3m ...(1)
Step 3:
To prove that P(k + 1) is true, i.e., to prove that `5^(2("k" + 1)) - 2^(2("k" + 1))` is divisible by 3.
Now, `5^(2("k" + 1)) - 2^(2("k" + 1))` = 52k+2 – 22k+2
= 52k .52 – 22k . 22
= (22k + 3m)25 – 22k . 4 ...[By (1)]
= 25(22k) + 75m – 4(22k)
= 21(22k) + 75m
= 3[7.22k + 25m]
∴ `(5^(2("k" + 1)) - 2^(2("k"+1)))/3` = 7.22k + 25m, where (7.22k + 25m) ∈ N
∴ `5^(2("k" + 1)) - 2^(2("k" + 1))` is divisible by 3
∴ P(k + 1) is true.
Step 4:
From all the above steps and by the principle of mathematical induction P(n) is true for all n ∈ N,
i.e., 52n – 22n is divisible by 3, for all n ∈ N.
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