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Sum

Find the term independent of x in the expansion of `(sqrt(x)/sqrt(3) + sqrt(3)/(2x^2))^10`.

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#### Solution

Let (r + 1)^{th} term be independent of x which is given by

T_{r+1} = `""^10"C"_r sqrt(x/3)^(10 - r) sqrt(3)^r/(2x^2)`

= `""^10"C"_r x^((10 - r)/2)/3 3^(r/2) 1/(2^r x^(2r))`

= `""^10"C"_r 3^(r/2 - (10 - r)/2) 2^(-r) x^((10 - r)/2 - 2r)`

Since the term is independent of x, we have

`(10 - r)/2 - 2r` = 0

⇒ r = 2

Hence 3^{rd} term is independent of x and its value is given by

T_{3} = `""^10"C"_2 (3^(-3))/4`

= `(10 xx 9)/(2 xx 1) xx 1/(9 xx 12)`

= `5/12`

Concept: Binomial Theorem for Positive Integral Indices

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