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Integral Calculus

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18 May 2011 22:39:45 IST

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P(n) = 5^n - 5 is divisible by 4. Prove it

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P(n) = 5^n - 5 is divisible by 4.Prove it

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NugoRama

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18 May 2011 23:00:53 IST

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P(n) = 5^n - 5

= 5[5^(n-1) - 1]

Now, a^n - 1 is divisible by (a-1)

(as a^n - b^n is divisible by (a-b))

(Use: a^n - b^n = (a-b) (a^(n-1) + a^(n-2)b + a^(n-3)b^2 + ... a^(n-1-k)b^k ... + b^(n-1)) )

Hence, P(n) is divisible by (5-1) ie 4.

Sushobhan Sen

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24 May 2011 18:11:07 IST

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Solve by mathematical induction.

P(1)=0, which is dividible by 4. So, P(1) is true ---- (1)

Let P(n) be true i.e., 5^n-5=4m, m is a natural number ---- (2) Hence, 5^n=4m+5 ---(3)

P(n+1) = 5^(n+1)-5

=5(5^n-1)

=5(4m+5-1) = 5(4m+4) = 4*(5m+5)=4p, p being some natural number, so 4p is divisible by 4

Hence, P(n+1) is true. --- (4)

From 1, 2 and 4 and by principle of finite mathematical induction, P(n) is true for all n.

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