# If the coefficients of x7 and x8 in 2+xn3 are equal, then n is ______. - Mathematics

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

• 56

• 55

• 45

• 15

#### Solution

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

Concept: Binomial Theorem for Positive Integral Indices
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#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 8 Binomial Theorem
Solved Examples | Q 17 | Page 139
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