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. - Mathematics

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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.

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

Concept: General and Middle Terms
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APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 8 Binomial Theorem
Exercise | Q 9 | Page 143

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