Question

find the sum by adding each term together. use the summation capabilities of a graphing utility to verify your result.
\\(\sum_{i = 2}^{5}(i - 1)^2 + (i + 1)^3\\)

Explanation:

Step1: Identify index values

The index \(i\) runs from 2 to 5, so \(i = 2, 3, 4, 5\).

Step2: Calculate term for \(i=2\)

Substitute \(i=2\) into the expression:
\[
(2-1)^2 + (2+1)^3 = 1^2 + 3^3 = 1 + 27 = 28
\]

Step3: Calculate term for \(i=3\)

Substitute \(i=3\) into the expression:
\[
(3-1)^2 + (3+1)^3 = 2^2 + 4^3 = 4 + 64 = 68
\]

Step4: Calculate term for \(i=4\)

Substitute \(i=4\) into the expression:
\[
(4-1)^2 + (4+1)^3 = 3^2 + 5^3 = 9 + 125 = 134
\]

Step5: Calculate term for \(i=5\)

Substitute \(i=5\) into the expression:
\[
(5-1)^2 + (5+1)^3 = 4^2 + 6^3 = 16 + 216 = 232
\]

Step6: Sum all calculated terms

Add the four results together:
\[
28 + 68 + 134 + 232 = 462
\]

Answer:

462