Evaluate the Summation sum from k=5 to 9 of sin(kpi)

Solution:

Evaluate the Summation $\sum _{k=5}^9\sin (k\pi )$

Step 1:
Write out the summation explicitly for each integer $k$ from 5 to 9.
$\sin (5\pi )+\sin (6\pi )+\sin (7\pi )+\sin (8\pi )+\sin (9\pi )$

Step 2:
Calculate each term in the sequence, noting that the sine of any integer multiple of $\pi$ is 0.
$0+0+0+0+0=0$

Knowledge Notes:

The problem involves evaluating a finite summation of sine functions where the argument of the sine function is an integer multiple of $\pi$. The relevant knowledge points for solving this problem include:

1. Summation Notation: The summation notation $\sum$ is used to denote the sum of a sequence of terms. The expression $\sum _{k=a}^{b}f(k)$ means that you should evaluate the function $f(k)$ for every integer $k$ from $a$ to $b$, and then add up all those values.

2. Sine Function Properties: The sine function has a period of $2\pi$, which means that $\sin (\theta +2\pi n)=\sin (\theta )$ for any integer $n$. Moreover, for any integer $m$, $\sin (m\pi )=0$ because sine is zero at multiples of $\pi$.

3. Simplification of Series: When evaluating a series, if each term in the series simplifies to zero, the sum of the series is also zero.

In this problem, we use the property of the sine function that $\sin (m\pi )=0$ for any integer $m$ to simplify each term in the summation to zero. Since all terms are zero, the sum is zero.