The sum of the series 2 × 1 × ²⁰C₄ − 3 × 2 × ²⁰C₅ + 4 × 3 × ²⁰C₆ − 5 × 4 × ²⁰C₇ + ⋯ + 18 × 17 × ²⁰C₂₀, is equal to:
❓ Question Evaluate the sum of the series: S = 2 × 1 × ( 20 4 ) − 3 × 2 × ( 20 5 ) + 4 × 3 × ( 20 6 ) − 5 × 4 × ( 20 7 ) + ⋯ + 18 × 17 × ( 20 20 ) . 🖼️ Question Image ✍️ Short Solution The given terms are of the form: ( − 1 ) r ( r + 1 ) r ( 20 r + 3 ) , r = 1 , 2 , … , 17. Simplify: ( r + 1 ) r ( 20 r + 3 ) = ( r + 3 ) ( r + 2 ) ( r + 1 ) r ⋅ 20 ! ( r + 3 ) ! ( 20 − r − 3 ) ! . Better trick: Use identity: r ( r + 1 ) ( 20 r + 3 ) = 20 × 19 × ( 18 r + 1 ) . So the sum becomes: S = ∑ r = 1 17 ( − 1 ) r ⋅ 20 × 19 × ( 18 r + 1 ) . Factor constants: S = 380 ∑ r = 1 17 ( − 1 ) r ( 18 r + 1 ) . Change index: let k = r + 1 k = r+1 . Then k = 2 , 3 , … , 18 k = 2,3,\ldots,18 . S = 380 ∑ k = 2 18 ( − 1 ) k − 1 ( 18 k ) . Separate sign: S = − 380 ∑ k = 2 18 ( − 1 ) k ( 18 k ) . But recall Binomial theorem: ∑ k = 0 18 ( − 1 ) k ( 18 k ) = ( 1 − 1 ) 18 = 0. So: ∑ k = 2 18 ( − 1 ) k ( 18 k ) = − [ ( 18 0 ) ( − 1 ) 0 + ( 18 1 ) ( − 1 ) 1 ] . \sum_{k=2}^{18}...