Question

In the binomial expansion of (1 + x)ⁿ, the coefficients of the 5^th, 6^th and 7^th terms are in AP. Find all values of n

Answer 1

Coefficient of T₅, T₆, T₇, are in A.P.
∴ ⁿC₄, ⁿC₅, ⁿC₆, are in A.P.

⇒ `("n"("n" - 1)("n" - 2)("n"- 3))/(4*3*2*1)`,

`("n"("n" - 1)("" - 2)("n" - 3)("n" - 4))/(5*4*3*2*1)`,

`("n"("n" - 1)("n"- 2)("n" - 3)("n" - 4)("n" - 5))/(6*5*4*3*1)` are in A.P.

Muliplying each erm by `(4*3*2*1)/("n"("n" - 1)("n" - 2)("n" - 3))`, we get

⇒ `1, ("n" - 4)/5, (("n" - 4)("n" - 5))/(6*5)` are in A.P.

⇒ `1, ("n" - 4)/5, ("n"^2 - 9"n" + 20)/30` arein A.P.

∴ `("n" - 4)/5 - 1 = ("n"^2 - 9"n" + 20)/30 - ("n" - 4)/5`

⇒ `("n" - 9)/5 =("n"^2 15"n" + 44)/30`

⇒ 6(n – 9) = n² – 15n + 44

⇒ n² – 21n + 98 = 0

⇒ (n – 1)(n – 14) = 0

∴ n = 7, 14