Advertisements
Advertisements
Question
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 ______.
Solution
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
APPEARS IN
RELATED QUESTIONS
Find the 11th term from the beginning and the 11th term from the end in the expansion of \[\left( 2x - \frac{1}{x^2} \right)^{25}\] .
Find the 7th term in the expansion of \[\left( 3 x^2 - \frac{1}{x^3} \right)^{10}\] .
Find the 7th term in the expansion of \[\left( \frac{4x}{5} + \frac{5}{2x} \right)^8\]
Find the 4th term from the beginning and 4th term from the end in the expansion of \[\left( x + \frac{2}{x} \right)^9\] .
Find the 4th term from the end in the expansion of \[\left( \frac{4x}{5} - \frac{5}{2x} \right)^8\] .
Find the 7th term from the end in the expansion of \[\left( 2 x^2 - \frac{3}{2x} \right)^8\] .
Find the sixth term in the expansion \[\left( y^\frac{1}{2} + x^\frac{1}{3} \right)^n\] , if the binomial coefficient of the third term from the end is 45.
if the seventh term from the beginning and end in the binomial expansion of \[\left( \sqrt[3]{2} + \frac{1}{\sqrt[3]{3}} \right)^n\] are equal, find n.
Write the number of terms in the expansion of \[\left( 2 + \sqrt{3}x \right)^{10} + \left( 2 - \sqrt{3}x \right)^{10}\] .
Write the number of terms in the expansion of \[\left( 2 + \sqrt{3}x \right)^{10} + \left( 2 - \sqrt{3}x \right)^{10}\] .
Write the number of terms in the expansion of \[\left( 1 - 3x + 3 x^2 - x^3 \right)^8\]
Which term is independent of x, in the expansion of \[\left( x - \frac{1}{3 x^2} \right)^9 ?\]
Write the number of terms in the expansion of \[\left[ \left( 2x + y^3 \right)^4 \right]^7\] .
Find the number of terms in the expansion of\[\left( a + b + c \right)^n\]
If rth term in the expansion of \[\left( 2 x^2 - \frac{1}{x} \right)^{12}\] is without x, then r is equal to
The coefficient of \[x^{- 3}\] in the expansion of \[\left( x - \frac{m}{x} \right)^{11}\] is
The coefficient of the term independent of x in the expansion of \[\left( ax + \frac{b}{x} \right)^{14}\] is
If the coefficients of 2nd, 3rd and 4th terms in the expansion of \[\left( 1 + x \right)^n , n \in N\] are in A.P., then n =
Constant term in the expansion of \[\left( x - \frac{1}{x} \right)^{10}\] is
