a_(n)=(3)/(5)[(5)/(n pi) sin (n pi x)/(5)]_(0)^5

To find the value of \( a_{n} \), we need to evaluate the expression inside the brackets first.<br /><br />The expression inside the brackets is \( \frac{5}{n \pi} \sin \frac{n \pi x}{5} \).<br /><br />We need to evaluate this expression at \( x = 0 \) and \( x = 5 \) and then subtract the two results.<br /><br />Let's start by evaluating the expression at \( x = 0 \):<br /><br />\( \frac{5}{n \pi} \sin \frac{n \pi \cdot 0}{5} = \frac{5}{n \pi} \sin 0 = \frac{5}{n \pi} \cdot 0 = 0 \)<br /><br />Now, let's evaluate the expression at \( x = 5 \):<br /><br />\( \frac{5}{n \pi} \sin \frac{n \pi \cdot 5}{5} = \frac{5}{n \pi} \sin n \pi = \frac{5}{n \pi} \cdot 0 = 0 \)<br /><br />Now, we can substitute these values back into the original expression:<br /><br />\( a_{n} = \frac{3}{5} \left[ \frac{5}{n \pi} \sin \frac{n \pi x}{5} \right]_{0}^{5} = \frac{3}{5} \left[ 0 - 0 \right] = \frac{3}{5} \cdot 0 = 0 \)<br /><br />Therefore, the value of \( a_{n} \) is 0.