Question

find the value of the summation.
\\(\sum\limits_{k=0}^{4} (k^2 + 4) = \square\\)
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Explanation:

Step1: Understand the summation

The summation is from \( k = 0 \) to \( k = 4 \) of \( (k^2 + 4) \). We need to substitute \( k = 0, 1, 2, 3, 4 \) into the expression and sum the results.

Step2: Substitute \( k = 0 \)

For \( k = 0 \): \( (0^2 + 4) = 0 + 4 = 4 \)

Step3: Substitute \( k = 1 \)

For \( k = 1 \): \( (1^2 + 4) = 1 + 4 = 5 \)

Step4: Substitute \( k = 2 \)

For \( k = 2 \): \( (2^2 + 4) = 4 + 4 = 8 \)

Step5: Substitute \( k = 3 \)

For \( k = 3 \): \( (3^2 + 4) = 9 + 4 = 13 \)

Step6: Substitute \( k = 4 \)

For \( k = 4 \): \( (4^2 + 4) = 16 + 4 = 20 \)

Step7: Sum all the results

Now, sum the values: \( 4 + 5 + 8 + 13 + 20 \)

First, \( 4 + 5 = 9 \)

Then, \( 9 + 8 = 17 \)

Next, \( 17 + 13 = 30 \)

Finally, \( 30 + 20 = 50 \)

Answer:

\( 50 \)