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Sum

Find the term independent of x in the expansion of `(3x - 2/x^2)^15`

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#### Solution

Given expression is `(3x - 2/x^2)^15`

General term `"T"_(r + 1) = ""^n"C"_r x^(n - r) y^r`

= `""^15"C"_r (3x)^(15 - r) (- 2/x^2)^r`

= `""^15"C"_r (3)^(15 - r) * x^(15 - r) (-2)^r * 1/x^(2r)`

= `""^15"C"_r (3)^(15 - r) * x^(15 - r - 2r) * (-2)^r`

= `""^15"C"_r (3)^(15 - r) * x^(15 - 3r) (-2)^r`

For getting a term independent of x

Put 15 – 3r = 0

⇒ r = 5

∴ The required term is `""^15"C"_5 (3)^(15 - 5) (-2)^5`

= `- ""^15"C"_3 (3)^10 (2)^5`

= `-(15 xx 14 xx 13 xx 12 xx 11)/(5 xx 4 xx 3 xx 2 xx 1) * (3)^10 (2)^5`

= `-7 xx 13 xx 3 xx 11 * 3(3)^10 (2)^5`

= – 3003 (3)^{10} (2)^{5}

Hence, the required term = –3003 (3)^{10} (2)^{5}

Concept: General and Middle Terms

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