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प्रश्न
If the seventh terms from the beginning and the end in the expansion of `(root(3)(2) + 1/(root(3)(3)))^n` are equal, then n equals ______.
उत्तर
If the seventh terms from the beginning and the end in the expansion of `(root(3)(2) + 1/(root(3)(3)))^n` are equal, then n equals 12.
Explanation:
The given expansion is `(root(3)(2) + 1/(root(3)(3)))^n`
∴ T7 = T6+1
= `""^n"C"_6 (2^(1/3))^(n - 6) * 1/((3^(1/3))^6`
= `""^n"C"_6 (2)^((n - 6)/3) * 1/(3)^2`
Now the T7 from the end = T7 from the beginning in `(1/(root(3)(2)) + root(3)(2))^n`.
∴ T7 = T6+1
= `""^n"C"_6 (1/(3^(1/3)))^(n - 6) * (2^(1/3))^6`
We get `""^n"C"_6 (2)^((n - 6)/3) * (1/3^2) = ""^n"C"_6 1/((n - 6)/3) * (2)^2`
⇒ `(2)^((n - 6)/3) * (3)^-2 = (3)^-((n - 5)/3) * (2)^2`
⇒ `(2)^((n - 6)/3 2) * (3)^(-2 + (n - 6)/3)` = 1
⇒ `2^((n - 12)/3) * (3)^((n - 12)/3)` = 1
⇒ `(6)^((n - 12)/3) = (6)^0`
⇒ `(n - 12)/3` = 0
⇒ n = 12
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