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Question
Find the constant term (term independent of x) in the expansion of `(2x^2 - 5/x)^9`
Solution
Let tr+1 be the constant term in the expansion of `(2x^2 - 5/x)^9`
We know that, in the expansion of (a+ b)n,
tr+1 = nCr an–r br
Here a = 2x2, b = `-5/x`, n = 9
∴ tr+1 = `""^9"C"_"r" (2x^2)^(9 - "r") ((-5)/x)^"r"`
= `""^9"C"_"r" 2^(9 - "r")*x^(18 - 2"r")*(-5)^"r"*x^(-"r")`
= `""^9"C"_"r" 2^(9 - "r")*(-5)^"r"*x^(18 - 3"r")`
But tr+1 is a constant term
∴ power of x = 0
∴ 18 – 3r = 0
∴ r = 6
∴ the constant term
= 9C6 29–6 · (– 5)6
= 9C3 · 23 · (– 5)6 ...[∵ nCr = nCn–r]
= `(9 xx 8 xx 7)/(1 xx 2 xx 3) xx 8 xx 15625`
= 10500000
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