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Find the sum of the following C_1 - 2C_2 + 3C_3 - ..... + n(-1)^(n-1) C_n |
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Answer» SOLUTION :L.H.S. : `C_0 + 3C_1 + 5C_2 + ... + (2N+1)C_n` `sum_(k=1)^n (2K+1) C_k` = `C_0 + sum_(k=1)^n(2k+1) "^nC_k `"^nC_0 + sum(k=1)^n 2k ^nC_k + sum(k=1)^n ^nC_k `"^nC_0 + 2sum_(k=1)^n n ^(n-1)C_(k-1) + (^nC_1 + ^nC_2 + ^nC_n)` `("^nC_0 + ^nC_1 + ^nC_2 + .... + ^nC_n) + 2n^(n-1)C_0 + ^(n-1)C_1 + ... + ^(n-1)C_(n-1) ) `2^n + 2n.2(n-1)` = `2^n + n.2^n` = `(n+1)2^n` R.H.S. |
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