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Question
Answer the following:
Find the constant term in the expansion of `((4x^2)/3 + 3/(2x))^9`
Solution
Let tr+1 be the constant term in the expansion of `((4x^2)/3 + 3/(2x))^9`.
We know that, in the expansion of (a+ b)n,
tr+1 = nCr, an–r br
Here a = `(4x^2)/3`, b = `3/(2x)`, n = 9
∴ tr+1 = `""^9"C"_"r" ((4x^2)/3)^(9-"r") (3/(2x))^"r"`
= `""^9"C"_"r" (4/3)^(9-"r") x^(18-2"r")*(3/2)^"r" x^(-"r")`
= `""^9"C"_"r" (4/3)^(9 - "r") (3/2)^"r" x^(18 - 3"r")`
But tr+1 is a constant term
∴ power of x = 0
∴ 18 – 3r = 0
∴ r = 6
∴ the constant term
= `""^9"C"_6 (4/3)^(9 - 6) (3/2)^6`
= `""^9"C"_3 (4/3)^3 (3/2)^6` ...[∵ nCr = nCn–r]
= `(9 xx 8 xx 7)/(1 xx 2 xx 3) xx 64/27 xx (27 xx 27)/64`
= 2268
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