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Question
Prove that 12C1−23C2+34C3−45C4+......+(−1)n+1⋅nn+1=1n+1
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Solution
L.H.S = (1−12)C1−(1−13)C2+(1−14)C3...+
(C1−C2+C3−C4+...)−(C12−C23+C34−...)
Now (C0−C1+C2−C3+...=0)
and C12−C23+C34....=C0−1n+1 by 3 last page
∴L.H.S=C0−(C0−1n+1)=1n+1
(C1−C2+C3−C4+...)−(C12−C23+C34−...)
Now (C0−C1+C2−C3+...=0)
and C12−C23+C34....=C0−1n+1 by 3 last page
∴L.H.S=C0−(C0−1n+1)=1n+1
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