The sum of the binomial coefficients in the expansion of (x-3/4 + ax5/4)n lies between 200 and 400 and the term independent of x equals 448. The value of a isa)1b)2c)1/2d)for no value of aCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Answer

Given:
The expansion of $(x-\frac{3}{4}ax^{\frac{5}{4}})^n$ has the following conditions:
1) The sum of the binomial coefficients is between 200 and 400.
2) The term independent of $x$ is 448.

To Find:
The value of $a$.

Solution:

1. Finding the Sum of Binomial Coefficients:
The sum of binomial coefficients in the expansion of $(x-\frac{3}{4}ax^{\frac{5}{4}})^n$ can be calculated using the binomial theorem.

The binomial theorem states that the $k^{th}$ term in the expansion of $(a+b)^n$ is given by:
$T_k = \binom{n}{k}a^{n-k}b^k$

In our case, $a = x$ and $b = -\frac{3}{4}ax^{\frac{5}{4}}$, and we need to find the sum of the binomial coefficients. Let's denote this sum as $S$.

$S = \binom{n}{0}a^n(-\frac{3}{4}ax^{\frac{5}{4}})^0 + \binom{n}{1}a^{n-1}(-\frac{3}{4}ax^{\frac{5}{4}})^1 + \binom{n}{2}a^{n-2}(-\frac{3}{4}ax^{\frac{5}{4}})^2 + ... + \binom{n}{n}(-\frac{3}{4}ax^{\frac{5}{4}})^n$

Since we are interested in the sum of binomial coefficients, we can ignore all terms with powers of $a$. This leaves us with:

$S = \binom{n}{0} + \binom{n}{1}(-\frac{3}{4}ax^{\frac{5}{4}}) + \binom{n}{2}(-\frac{3}{4}ax^{\frac{5}{4}})^2 + ... + \binom{n}{n}(-\frac{3}{4}ax^{\frac{5}{4}})^n$

Notice that this is the expansion of $(1 - \frac{3}{4}ax^{\frac{5}{4}})^n$.

So, $S = (1 - \frac{3}{4}ax^{\frac{5}{4}})^n$

Given that the sum of binomial coefficients is between 200 and 400, we have:

$200 < (1='' -='' \frac{3}{4}ax^{\frac{5}{4}})^n='' />< />

2. Finding the Term Independent of x:
The term independent of $x$ is obtained when all the powers of $x$ cancel out in the expansion. This occurs when the exponent of $x$ in each term is zero. In other words, the exponent of $x$ in each term is a multiple of $\frac{5}{4}$.

The term independent of $x$ in the expansion of $(x-\frac{3}{4}ax^{\frac{5}{4}})^

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