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-7 , where 
a is a constant. Find the value of 
a
It is given that the constant term in the expansion of 
(2+(1)/(x))^(2)(1-3x)^(n^(2)) is 67 , where 
n is a positive integer. Find the value of 
n.

(4+(4)/(x)+(1)/(x^(2)))[C_(0)^(n)+C_(d)^(n)(-3x)+C_(2)^(n)9:}

7-7 , where a a is a constant. Find the value of a a \newlineIt is given that the constant term in the expansion of (2+1x)2(13x)n2 \left(2+\frac{1}{x}\right)^{2}(1-3 x)^{n^{2}} is 6767 , where n n is a positive integer. Find the value of n n .\newline(4+4x+1x2)[C0n+Cdn(3x)+C2n9 \left(4+\frac{4}{x}+\frac{1}{x^{2}}\right)\left[C_{0}^{n}+C_{d}^{n}(-3 x)+C_{2}^{n} 9\right.

Full solution

Q. 7-7 , where a a is a constant. Find the value of a a \newlineIt is given that the constant term in the expansion of (2+1x)2(13x)n2 \left(2+\frac{1}{x}\right)^{2}(1-3 x)^{n^{2}} is 6767 , where n n is a positive integer. Find the value of n n .\newline(4+4x+1x2)[C0n+Cdn(3x)+C2n9 \left(4+\frac{4}{x}+\frac{1}{x^{2}}\right)\left[C_{0}^{n}+C_{d}^{n}(-3 x)+C_{2}^{n} 9\right.
  1. Simplify Expression: Identify the general form of the expression and simplify the first part.\newline(2+1x)2=4+4x+1x2(2 + \frac{1}{x})^2 = 4 + \frac{4}{x} + \frac{1}{x^2}
  2. Expand Using Binomial Theorem: Expand the second part using the binomial theorem.\newline(13x)n2=C0n(3x)0+C1n(3x)1+C2n(3x)2+(1 - 3x)^{n^2} = C_0^n(-3x)^0 + C_1^n(-3x)^1 + C_2^n(-3x)^2 + \ldots
  3. Combine and Find Constant Term: Combine the expansions and look for the constant term.\newline(4+4x+1x2)(C0n+C1n(3x)+C2n(9x2)+)(4 + \frac{4}{x} + \frac{1}{x^2})(C_0^n + C_1^n(-3x) + C_2^n(9x^2) + \ldots)\newlineTo find the constant term, match powers of x to zero:\newline- From 4C1n(3x)4\cdot C_1^n(-3x), x-term coefficient is 12C1n-12C_1^n\newline- From 1x29C2nx2\frac{1}{x^2} \cdot 9C_2^n x^2, constant term is 9C2n9C_2^n

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