Question

The largest value of x for which given expression $\sum_{k=0}^{4} (\frac{5^{4-k}}{(4-k)!}) (\frac{x^k}{k!}) = \frac{1}{384}$ holds true is k then $[k] + 5$ is equal to where [.] represents greatest integer function

Answer 1

0

Answer 2

The given expression is $\sum_{k=0}^{4} (\frac{5^{4-k}}{(4-k)!}) (\frac{x^k}{k!})$. Let's analyze the general term of the summation: $\frac{5^{4-k}}{(4-k)!} \frac{x^k}{k!}$. We can multiply and divide by $4!$ to relate this to binomial coefficients: $\frac{1}{4!} \frac{4!}{(4-k)!k!} 5^{4-k} x^k = \frac{1}{4!} \binom{4}{k} 5^{4-k} x^k$.

So the summation can be written as: $\sum_{k=0}^{4} \frac{1}{4!} \binom{4}{k} 5^{4-k} x^k = \frac{1}{4!} \sum_{k=0}^{4} \binom{4}{k} 5^{4-k} x^k$.

Recall the binomial theorem: $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$. In our case, $n=4$, $a=5$, and $b=x$. So, $\sum_{k=0}^{4} \binom{4}{k} 5^{4-k} x^k = (5+x)^4$.

The given expression is therefore equal to $\frac{1}{4!} (5+x)^4$. We are given that this expression equals $\frac{1}{384}$. $\frac{1}{4!} (5+x)^4 = \frac{1}{384}$.

Calculate $4!$: $4! = 4 \times 3 \times 2 \times 1 = 24$. The equation becomes: $\frac{1}{24} (5+x)^4 = \frac{1}{384}$.

Multiply both sides by 24: $(5+x)^4 = \frac{24}{384}$.

Simplify the fraction $\frac{24}{384}$: $\frac{24}{384} = \frac{24}{24 \times 16} = \frac{1}{16}$.

So, the equation is $(5+x)^4 = \frac{1}{16}$. To solve for $5+x$, we take the fourth root of both sides: $5+x = \pm \sqrt[4]{\frac{1}{16}}$. $\sqrt[4]{\frac{1}{16}} = \sqrt[4]{(\frac{1}{2})^4} = \frac{1}{2}$. So, $5+x = \pm \frac{1}{2}$.

This gives two possible cases for $5+x$:

Case 1: $5+x = \frac{1}{2}$ $x = \frac{1}{2} - 5 = \frac{1}{2} - \frac{10}{2} = -\frac{9}{2} = -4.5$.

Case 2: $5+x = -\frac{1}{2}$ $x = -\frac{1}{2} - 5 = -\frac{1}{2} - \frac{10}{2} = -\frac{11}{2} = -5.5$.

The possible values of $x$ are $-4.5$ and $-5.5$. We are asked for the largest value of $x$. Comparing the two values, $-4.5$ is larger than $-5.5$. So, the largest value of $x$ is $-4.5$.

The question states that the largest value of $x$ is $k$. Thus, $k = -4.5$.

We need to calculate $[k] + 5$, where $[.]$ represents the greatest integer function. $[k] = [-4.5]$. The greatest integer less than or equal to $-4.5$ is $-5$. So, $[-4.5] = -5$.

Finally, $[k] + 5 = -5 + 5 = 0$.