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Question
Find the value of r, if the coefficients of (2r + 4)th and (r – 2)th terms in the expansion of (1 + x)18 are equal.
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Solution
General Term `"T"("r" + 1) = ""^"n""C"_"r" x^("n" - "r") y^"r"`
For coefficient of (2r + 4)th term, we have
`"T"_(2r + 4) = "T"_(2r + 3 + 1)`
= `""^18"C"_(2r + 3) (1)^(18 - 2r - 3) * x^(2r + 3)`
∴ Coefficient of (2r + 4)th term = `""^18"C"_(2r + 3)`
Similarly, `"T"_(r - 2) = "T"_(r - 3 + 1)`
= `""^18"C"_(r - 3) (1)^(18 - r + 3) * x^(r - 3)`
∴ Coefficient of (r – 2)th term = `""^18"C"_(r - 3)`
As per the condition of the questions,
We have `""^18"C"_(2r + 3) = ""^18"C"_(r - 3)`
⇒ 2r + 3 + r – 3 = 18
⇒ 3r = 18
⇒ r = 6
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