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Question

Prove that
12+32+52+...+(2n1)2=n3(2n1)(2n+1)

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Solution

Let p(n)
=12+32+....+(2n1)2=n(2n1)(2n+1)3
for n=1
LHS=12=1
RHS=1(1)(3)3=1
so, p(n) is true for n=1
Let p(k) be true
12+32+....+(2k1)2=(2k1)(2k+1)3(1)
for p (k + 1)
12+22+...+(2(k+1)1)2=(k+1)(2(k+1)1)3(2(k+1)+1)
=12+32+...+(2k+1)2=(k+1)2k+13(2k+3)(2)
Form (1)
adding (2k+1)2 on both sides.
=12+...+(2k+1)2=k(2k1)(2k+1)3+(2k+1)2
=(2k+1)[k(2k1)+3(2k+1)3]
=(2k+1)[2k2k+6k+33]
=(2k+1)(2k2+5k+33)
=(k+1)(2k+1)(2k+3)3
so, using mathematical induction
12+...(2n+1)2=n(2n1)(2n+1)3

1206887_1296616_ans_a400fbaf8a1e492d9eac4b55aeb330f1.jpg

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