Advertisement Remove all ads

Advertisement Remove all ads

Advertisement Remove all ads

MCQ

Fill in the Blanks

If the coefficients of x^{7} and x^{8} in `2 + x^n/3` are equal, then n is ______.

#### Options

56

55

45

15

Advertisement Remove all ads

#### Solution

If the coefficients of x^{7} and x^{8} 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, T_{8} = `""^n"C"_7 (2)^(n - 7) (x/3)^7`

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

And T_{9} = `""^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 x^{7} = coefficient x^{8})

⇒ `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

Concept: Binomial Theorem for Positive Integral Indices

Is there an error in this question or solution?