Prove that : (n − 1)². C 1 + (n − 3)². C 3 + (n − 5)². C 5 +..... = n (n + 1)2 n − 3
Text Solution
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(n – 1) 2 n C 1 + (n – 3) 2 n C 3 + (n – 5) 2 n C 5 + ......
= n 2 ( n C 1 + n C 3 + n C 5 + ........) – 2n ( n C 1 + 3 n C 3 + 5 n C 5 + .....) + ( n C 1 + 9 n C 3 + 25 n C 5 + .....)
= n 2 . 2 n – 1 – 2n 2 ( n–1 C 0 + n–1 C 2 + n–1 C 4 + ....) + n( n–1 C 0 + 3 n–1 C 2 + 5 n–1 C 4 + .....)
= n 2 . 2 n – 1 – 2n 2 . (2 n – 2 ) + n( n–1 C 0 + n–1 C 2 + n–1 C 4 + ....) + n(2 n–1 C 2 + 4 n–1 C 4 + 6 n–1 C 6 + ......)
= n 2 . 2 n – 1 – n 2 . 2 n – 1 + n . 2 n – 2 + n(n – 1) ( n – 2 C 1 + n – 2 C 3 + n – 2 C 5 + ...)
= n . 2 n – 2 + n(n – 1) 2 n – 3
= n(n + 1) 2 n – 3 .
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