: Sum of the series $$S=3$$−$$1$$ $$10C0$$−$$32$$ $$10C1+32$$ $$10C2$$−$$33$$ $$10C3+$$....…$$+310$$ $$10C10$$ is

We have $$S=3$$−$$1$$ $$10C0$$−$$10C1(3)+10C232$$−$$10C333+$$….....$$+10C10310$$ $$=13(1$$−$$3)10=13210$$

Binomial theorem for positive integral Index

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Question

: Sum of the series $S = 3^{- 1} [(^{10} C_{0}) - 3^{2} (^{10} C_{1}) + 3^{2} (^{10} C_{2}) - 3^{3} (^{10} C_{3}) + . . . . \dots + 3^{10} (^{10} C_{10})]$ is

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Solution

We have
$\begin{array}{l} S = 3^{- 1} [^{10} C_{0} -^{10} C_{1} (3) +^{10} C_{2} (3^{2}) -^{10} C_{3} (3^{3}) + \dots . . . . . +^{10} C_{10} (3^{10})] \\ = \frac{1}{3} (1 - 3)^{10} = \frac{1}{3} (2^{10}) \end{array}$

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