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Question
Answer the following:
Show that there is no constant term in the expansion of `(2x - x^2/4)^9`
Solution
Let, if possible, tr+1 is the constant term in the expansion of `(2x - x^2/4)^9`.
We know that, in the expansion of (a+ b)n,
tr+1 = nCr, an–r br
Here a = 2x, b = `- x^2/4`, n = 9
tr+1 = `""^9"C"_"r" (2x)^(9 - "r") (-x^2/4)^"r"`
= `""^9"C"_"r" 2^(9 - "r") * x^(9 - "r")*(-1/4)^"r" x^(2"r")`
= `""^9"C"_"r"* 2^(9 - "r")* (-1/4)^"r"* x^(9 + "r")`
But tr+1 is a constant term
∴ power of x = 0
∴ 9 + r = 0
∴ r = – 9
This is not possible, because r ∈ N.
Hence, there is no constant term in the expansion of `(2x - x^2/4)^9`.
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