Prove that n C r + 2 n +1 C r + 3 n+2 C r +....... + (n + 1…

Maths Binomial Theorem & Mathematical Induction Properties of Binomial Coefficients Subjective Type

Published on: August 13, 2026

Prove that n C r + 2 n +1 C r + 3 n+2 C r +....... + (n + 1) 2n C r = n C r+2 + (n + 1) 2n + 1 C r+1 – 2n+1 C r+2

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Let n C r + 2. n +1 C r + 3. n+2 C r +.......= S

S = co-efficient of x r in (1+x) n + 2.(1+x) n+1 + 3.(1+x) n+2 + ................

Let S' = (1+x) n + 2(1+x) n+1 + 3(1+x) n+2 + .....+ (n+1)(1+x) 2n ...............

(1+x) S' =(1+x) n+1 + 2(1+x) n+2 + ...+ (n+1)(1+x) 2n+1 ...............

– :

–x S' = (1+x) n + (1+x) n+1 + .....+ (1+x) 2n –(n+1) (1+x) 2n+1

⇒ – x S' = (1+x) n – (n+1)(1+x) 2n+1 ⇒ S' = +

Now S = co-efficient of x r in S'

= – 2n+1 C r+2 + n C r+2 + (n+1) 2n+1 C r+1

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