If in the expansion of (2^x+(1)/(4^x))^n, (T3)/(T2)=7 and the sum of the coefficients of 2 nd an

step 1 in this problem since t3 by t2 is equals to 7 we have nc2 2 power of x power n minus 2 into 4 po

If in the expansion of (2^x+(1)/(4^x))^n, (T3)/(T2)=7 and...

If in the expansion of $\left(2^{x}+\frac{1}{4^{x}}\right)^{n}, \frac{T_{3}}{T_{2}}=7$ and the sum of the coefficients of 2 nd and 3 rd terms is 36 , then the value of $x$ is
(A) $-\frac{1}{3}$
(B) $-\frac{1}{2}$
(C) $\frac{1}{3}$
(D) $\frac{1}{2}$

Key Concepts

Binomial Theorem

The Binomial Theorem provides a method to expand expressions raised to a positive integer power, specifically of the form (a + b)^n. It states that this expression can be expanded as a sum of terms involving binomial coefficients, where each term consists of a binomial coefficient multiplied by a power of a and a power of b. This theorem is fundamental in understanding how each term in a binomial expansion is formed.

General Term in a Binomial Expansion

The general term in a binomial expansion can be expressed as T_{k+1} = (n choose k) * a^(n-k) * b^(k), where k ranges from 0 to n. This formulation is important as it allows for a systematic approach to identify and manipulate specific terms within the expansion, particularly when comparing or equating certain terms, as is common in advanced algebra problems.

Ratio of Consecutive Terms

The ratio of consecutive terms in a binomial expansion is often used to establish relationships between the terms, especially when parameters of the expansion need to be solved. This is done by dividing one term by the previous term, which typically simplifies to an expression involving the binomial coefficients and the powers of the components a and b. This technique provides a powerful tool in problems where conditions on the terms’ ratios are given.

Exponential Expressions in Binomial Expansion

Exponential expressions within the context of binomial expansion involve terms where the components a and b might be exponential functions. Managing these exponents effectively often requires using properties of exponents to simplify or combine terms, which is crucial when the problem involves determining unknown exponent parameters based on given conditions about the terms.

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