Question

If the coefficients of x⁷ and x⁸ in `2 + x^n/3` are equal, then n is ______.

Options

• 56

• 55

• 45

• 15

Answer 1

If the coefficients of x⁷ and x⁸ in `2 + x^"n"/3` are equal, then n is 55.

Explanation:

Since `"T"_("r" + 1) = ""^"n""C"_"r"  "a"^("n" - "r")  x^"r"` in expansion of (a + x)ⁿ

Therefore, T₈ = `""^"n""C"_7 (2)^("n" - 7)  (x/3)^7`

= `""^"n""C"_7  (2^("n" - 7))/3^7  x^7`

And T₉ = `""^"n""C"_8  (2)^("n" - 8)  (x/3)^8`

= `""^"n""C"_8  (2^("n" - 8))/3^8  x^8`

Therefore, `""^"n""C"_7  (2^("n" - 7))/3^7`

= `""^"n""C"_8 (2^("n" - 8))/3^8`   ....(Since it is given that coefficient of x⁷ = coefficient x⁸)

⇒ `"n"/((7)("n" - 7)) xx (8("n" - 8))/"n" = (2^("n" - 8))/3^8 * 3^7/(2^("n" - 7))`

⇒ `8/("n" - 7) = 1/6`

⇒ n = 55