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1 2 +3 2 +5 2 ….+(2n-1) 2 =1/3 n(4n 2 -1) by induction

12+32+52….+(2n-1)2=1/3 n(4n2-1) by induction

Grade:11

1 Answers

Ravi
askIITians Faculty 69 Points
6 years ago
Check for n=1...(True)
Let it be true for n=k
i.e.
12+32+52….+(2k-1)2= k(4k2-1)/3 ...(i)

To prove: It is true for n=k+1

Soln.
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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