Question

Prove that (n^2+n) is divisible by 2 for any positive integer n.​

Answer 1

Answer:

Let, f(n)=n2−n; n∈N

Now, f(1)=0 is divisible by ′2′

Again f(2)=2²−2=2; divisible by ′2′

So, f(1),f(2) are true.

Let us assume that f(k) is true i.e.,

f(k)=k²−k=2k1​;   k1​∈z

Now,

f(k+1)=(k+1)2−(k+1)

=k(k+1)

=k²+k

=k²−k+2k

=2k1​+2k [using (1)]

=2(k1​+k);

So, f(x+1) is divisible by 2.

Now, by principle of mathematical induction we have,

f(x) is true ∀n∈N.

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