Question

plsss someone solve this question

its urgent​

Answer 1

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

Let P(n): 1² + 3² + 5² + . .... ..+(2n-1)² = n(2n-1)(2n+1)/3

step1:- for n = 1

P(1): 1² = 1(2-1)(2+1)/3 = 3/3 = 1

it's true.

step2:- for n = K

P(K): 1² + 3² + 5² +.....+(2k-1)² = K(2K-1)(2K-1)/3 _________(1)

step3:- for n= k+1

P(K+1): 1² + 3² + 5² + .......+(2k+1)² = (k+1)(2K+1)(2k+3)/3

from eqn (1)

1² + 3² + 5² + .. ......+ (2K -1)² = K(2K-1)(2K+1)/3

add ( 2K+1)² both sides,

1² + 3² + 5² + ......+ (2k-1)² + (2K+1)² = K(2k-1)(2k+1)/3 + (2K+1)²

= (2k+1)/3{2k² - K + 6K + 3}

= (2K+1)(2K² + 5K + 3)/3

= (2K+1)(2K+3)(K+1)/3

= [(k+1){2(k+1)-1}{2(K+1)+1}]/3

hence, p(k+1) is true when P(K) is true , from the principle of mathematical induction , statement is true for all real numbers.