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Sum

Determine whether the expansion of `(x^2 - 2/x)^18` will contain a term containing x^{10}?

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#### Solution

Let T_{r+1} contain x^{10}.

Then T_{r+1} = `""^18"C"_r (x^2)^(18 - r) (-2^r)/x`

= `""^18"C"_r x^(36 - 2r) (-1)^r * 2^r x^(-r)`

= `(-1)^r 2^r ""^18"C"_r x^(36 - 3r)`

Thus, 36 – 3r = 10

i.e., r = `36/3`

Since r is a fraction, the given expansion cannot have a term containing x^{10}.

Concept: Binomial Theorem for Positive Integral Indices

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