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Find the sum 3^n C0-8^n C1+13^n C2-18^n ...

Find the sum `3^n C_0-8^n C_1+13^n C_2-18^n C_3+dot +.....+(n+1)terms`

Text Solution

Verified by Experts

The general term of the series is `T_(r) = (-1)^(r ) (3+5r).^(n)C_(r )`
where `r = 0, 1, 2, "…..", n`. Therefore, sum of the series is given by
`S=underset(r=0)overset(n)sum(-1)^(r)(3+5r).^(n)C_(r)`
`=3(underset(r=0)overset(n)sum(-1)^(r).^(n)C_(r))+5(underset(r=1)overset(n)sum(-1)^(r ) n .^(n-1)C_(r-1))`
`= 3(underset(r=0)overset(n)sum(-1)^(r).^(n)C_(r))-5(underset(r=1)overset(n)sum(-1)^(r ) .^(n-1)C_(r-1))`
`= 3(1-1)^(n) - 5n(1-1)^(n-1)`
`= 0`
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