Question

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

Options

• 56

• 55

• 45

• 15

Answer 1

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