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  • 4n – 1 is divisible by 3, for each natural number n.

Prove each of the statements in Exercises 3 to 16 by the Principle of Mathematical Induction.

4^n -1 is divisible by 3, for each natural number n.


 

Answers (1)

P (n) = 4^n -1 is divisible by 3          …….. (Given)

Now, we’ll substitute different values for n,

P (0) = 4^0 - 1 = 0, viz. divisible by 3

P (1) = 4^1 - 1 = 3, divisible by 3.

P (2) = 4^2 - 1 = 15, divisible by 3

Now, let P (k) = 4^k - 1, is divisible by 3

Thus, 4^k - 1 = 3x

We also get that,

P (k+1) = 4^{k+1} - 1 = 4(3x+1) - 1

                      = 12x + 3, is divisible by 3

Thus, P (k+1) is also true,

Hence, by mathematical induction,

For each natural no. n it is true that, P (n) = 4n -1 is divisible by 3

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