If \(I_{n}=(\pi / 4) \int_{0} \tan ^{n} x d x\) then \({ }^{5} \sum_{r=1}\left[I /\left(I_{r}+I_{r+2}\right)\right]\) is equal to...... (a) 5 (b) 10 (c) 15 (d) 20

The notation \(I_n\) represents the integral: \[ I_n = \frac{\pi}{4} \int_{0} \tan^n x \, dx \] The notation \({ }^5 \sum_{r=1}\) signifies the summation of a given expression, with r going from 1 to 5. Now, we'll work further on simplifying the given expression and finding the value of \({ }^5 \sum_{r=1}\left[ \frac{I}{I_{r} + I_{r+2}}\right]\).

We know that \(I_r = \frac{\pi}{4} \int_{0} \tan^r x \, dx\) and \(I_{r+2} = \frac{\pi}{4} \int_{0} \tan^{r+2} x \, dx\). Adding these two quantities, we get: \[I_r + I_{r+2} = \frac{\pi}{4} \int_{0} \tan^r x \, dx + \frac{\pi}{4} \int_{0} \tan^{r+2} x \, dx = \frac{\pi}{4} \int_{0} (\tan^r x + \tan^{r+2} x) dx \] Now, the given expression becomes: \[{}^5 \sum_{r=1}\left[ \frac{I}{I_{r} + I_{r+2}}\right] = \sum_{r=1}^{5} \frac{\frac{\pi}{4} \int_{0} \tan^r x \, dx}{\frac{\pi}{4} \int_{0} (\tan^r x + \tan^{r+2} x) dx} \] The \(\frac{\pi}{4}\) term cancels out: \[ \sum_{r=1}^{5} \frac{\int_{0} \tan^r x \, dx}{\int_{0} (\tan^r x + \tan^{r+2} x) dx} \]

Let's observe the expression inside the summation: \[\frac{\int_{0} \tan^r x \, dx}{\int_{0} (\tan^r x + \tan^{r+2} x) dx}\] Notice that the denominator is just the sum of \(\int_{0} \tan^r x \, dx\) with a higher power of \(x\), which means the denominator will always be larger than the numerator. Specifically, we can write this expression as: \[\frac{1}{1+\int_{0} \tan^{r+2} x \, dx}\] With that, let us compute the summation: \[ S = \sum_{r=1}^{5} \frac{1}{1+\int_{0} \tan^{r+2} x \, dx}\] Even though we cannot find a closed form for the integrals in the expression, we can see that the summation converges to a finite value because each term in the summation is a fraction with numerator 1 and a denominator greater than 1. However, without any additional information or simplification, we are missing the necessary information to derive an exact numerical value. We cannot select an answer from the given choices (a, b, c, or d) using the information provided.