Question

find the indicated sum.
\\(\sum\limits_{i=1}^{4} i(i + 2)\\)
\\(\sum\limits_{i=1}^{4} i(i + 2) = \square\\) (simplify your answer.)

Explanation:

Step1: Expand the term inside the sum

$i(i+2) = i^2 + 2i$

Step2: Split the sum into two parts

$\sum_{i=1}^4 (i^2 + 2i) = \sum_{i=1}^4 i^2 + 2\sum_{i=1}^4 i$

Step3: Calculate the sum of squares

$\sum_{i=1}^4 i^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$

Step4: Calculate the sum of integers

$\sum_{i=1}^4 i = 1 + 2 + 3 + 4 = 10$; $2\sum_{i=1}^4 i = 2×10 = 20$

Step5: Add the two results

$30 + 20 = 50$

Answer:

50