n(n+1)(n+5) is multiple of 3
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Posted by Meenu Rani 7 years, 10 months ago
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Chandrashekhar Wankhede 7 years, 10 months ago
To verify : n(n+1)(n+5) is divisible by 3 or not
Let us use Principle of Induction
Let f(n) = n(n+1)(n+5) = n3+6n2+5n
Let us verify for n= 1: f(1) = 12 = 3*4 ; Hence f(n) is divisible by 3 for n = 1
Let us assume that f(n) is divisible by 3 for any natural number n = a
That is f(a) = a3+6a2+5a = 3 (m) ;
Let us Prove that f(a+1) is also divisible by 3 if f(a) is divisible by 3
f(a+1) = (a+1)3+6(a+1)2+5(a+1) = a3+9a2+20a+12 = (a3+6a2+5a)+3a2+15a+12 = 3m+3a2+15a+12
= 3(m+a2+5a+4); Hence f(a+1) is also divisible by 3.
Hence it is TRUE that n(n+1)(n+5) is divisible by 3
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