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.

Answer 1

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