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Question
If the coefficients of x7 and x8 in `2 + x^n/3` are equal, then n is ______.
Options
56
55
45
15
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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
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