Evaluate the Summation sum from j=1 to 5 of sin((jpi)/2)+3j-4
The problem you've presented is asking for the evaluation of a finite summation. The summation involves an expression that combines a trigonometric function (sine), a linear term, and a constant. Specifically, you're being asked to calculate the sum of the given expression as the index variable j ranges from 1 to 5. For each value of j, you would plug it into the sine function sinusoidal component which is sin((j*pi)/2), add three times the value of j (3j), and then subtract 4. After computing this expression for each integer value of j from 1 through 5, you would sum all of those results to get the final answer.
$\sum_{j = 1}^{5} sin \left(\right. \frac{j \pi}{2} \left.\right) + 3 j - 4$
Answer
Solution:
Step 1:
Write out the summation for each individual term where $j$ ranges from 1 to 5.
$ \sin\left(\frac{1\pi}{2}\right) + 3 \cdot 1 - 4 + \sin\left(\frac{2\pi}{2}\right) + 3 \cdot 2 - 4 + \sin\left(\frac{3\pi}{2}\right) + 3 \cdot 3 - 4 + \ldots + \sin\left(\frac{5\pi}{2}\right) + 3 \cdot 5 - 4 $
Step 2:
Calculate the sum of the series.
$ \sum = 26 $
Knowledge Notes:
The problem involves evaluating a finite summation of a trigonometric function combined with a linear function of $j$. The steps to solve this problem include:
Expansion of the Series: The summation is expanded by substituting the values of $j$ from 1 to 5 into the given expression $\sin\left(\frac{j\pi}{2}\right) + 3j - 4$.
Trigonometric Values: Recognize the pattern in the trigonometric function $\sin\left(\frac{j\pi}{2}\right)$, which has specific values for $j=1,2,3,4,5$. For example, $\sin\left(\frac{\pi}{2}\right) = 1$, $\sin(\pi) = 0$, $\sin\left(\frac{3\pi}{2}\right) = -1$, and so on.
Arithmetic Calculation: After expanding the series and substituting the trigonometric values, the next step is to perform the arithmetic operations, adding and subtracting the terms.
Summation: Finally, sum all the terms to get the result of the summation.
In this problem, the trigonometric function $\sin\left(\frac{j\pi}{2}\right)$ will yield a repeating pattern of values as $j$ increases, and the linear function $3j - 4$ will increase linearly with $j$. The final summation is the result of combining these two patterns for each value of $j$ from 1 to 5.
