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Check for n=1...(True)Let it be true for n=ki.e.12+32+52….+(2k-1)2= k(4k2-1)/3 ...(i)To prove: It is true for n=k+1Soln.12+32+52….+[2(k+1)-1]2= k(4k2-1)/3+ [2k+1]2 ...(from i)= k(2k+1)(2k-1)/3+ (2k+1)2=(2k+1)[k(2k-1)/3-(2k+1)]=(2k+1)(2k+3)(k+1)/3=(k+1)[2(k+1)+1][2(k+1)-1]/3=(k+1)[4(k+1)2-1]/3
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