Question

1²+3²+5²+...+(2n-1)²=[n(2n-1)(2n+1)]/3
Mathematical induction
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Answer 1

Given : 1²+3²+5²+...+(2n-1)²=[n(2n-1)(2n+1)]/3

To Find :  Prove using Mathematical induction

Solution:

P(n) = 1²+3²+5²+...+(2n-1)²=[n(2n-1)(2n+1)]/3

P(1)    n = 1

LHS = 1² = 1

RHS = [1(2-1)(2+1)]/3  = 1

Satisfied

Lets assume n = k satisfy

=> p(k)  = 1²+3²+5²+...+(2k-1)²=[k(2k-1)(2k+1)]/3

n = k + 1

=> p(k + 1)  =    1²+3²+5²+...+(2k-1)² + (2k + 1)²

= [k(2k-1)(2k+1)]/3 + (2k + 1)²

=> [(2k + 1)/3]  [k(2k-1)  + 3(2k + 1)]

= [(2k + 1)/3]  [2k² - k + 6k +  3]

= [(2k + 1)/3]  [2k²  + 5k + 3]

= [(2k + 1)/3]  [2k²  + 2k + 3k + 3]

= [(2k + 1)/3]  [2k(k + 1) + 3(k + 1)]

= [(2k + 1)/3]  [(k + 1)(2k + 3)]

=  [(k + 1)[(2k + 1)(2k + 3)]/3

which was required

Hence proved

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Answer 2

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