Answer the following: Prove by method of induction 52n − 22n is divisible by 3, for all n ∈ N

Question

Answer the following:

Prove by method of induction 5²ⁿ − 2²ⁿ is divisible by 3, for all n ∈ N

Sum

Solution

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. 5^2k – 2^2k is divisible by 3.

∴ `(5^(2"k") - 2^(2"k"))/3` = m (Say), whre m ∈ N

∴ 5^2k – 2^2k = 3m

∴ 5^2k = 2^2k + 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))` = 5^2k+2 – 2^2k+2

= 5^2k .5² – 2^2k . 2²

= (2^2k + 3m)25 – 2^2k . 4 ...[By (1)]

= 25(2^2k) + 75m – 4(2^2k)

= 21(2^2k) + 75m

= 3[7.2^2k + 25m]

∴ `(5^(2("k" + 1)) - 2^(2("k"+1)))/3` = 7.2^2k + 25m, where (7.2^2k + 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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Principle of Mathematical Induction

Is there an error in this question or solution?

Q II. (11) (iii)Q II. (11) (ii)Q II. (12)

Chapter 4: Methods of Induction and Binomial Theorem - Miscellaneous Exercise 4.2 [Page 86]

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Balbharati Mathematics and Statistics (Arts and Science) Part 2 [English] Standard 11 Maharashtra State Board

Chapter 4 Methods of Induction and Binomial Theorem
Miscellaneous Exercise 4.2 | Q II. (11) (iii) | Page 86

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