Express the Sum in Terms of N $\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } + 5 k + 5 \right)$

question 11

Multiple Choice

Express the sum in terms of n. $\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } + 5 k + 5 \right)$ (Hint: Use the theorem on sums to write the sum as $\sum _ { k = 1 } ^ { n } k ^ { 2 } + 5 \sum _ { k = 1 } ^ { n } k + \sum _ { k = 1 } ^ { n } 5$ . Next, use the following formulas $$\begin{array} { l } 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + n ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } \\1 + 2 + 3 + \ldots + n = \frac { n ( n + 1 ) } { 2 }\end{array}$$ and the theorem on the sum of a constant.)

• A) \(\frac { n ^ { 3 } - 9 n ^ { 2 } + 23 n } { 3 }\)
• B) \(\frac { n ^ { 3 } + 9 n ^ { 2 } - 23 n } { 3 }\)
• C) \[\frac { n ^ { 3 } + 9 n ^ { 2 } + 23 n } { 3 }\]
• D) \[\frac { n ^ { 3 } - 9 n ^ { 2 } - 23 n } { 3 }\]
• E) \(\frac { n ^ { 3 } + 3 n ^ { 2 } + 46 n } { 3 }\)

---

Definitions:

Injured Third Party

An individual or entity that suffers harm or damage due to the actions or the contractual relationship between two other parties.

Partnership

A legal form of business operation between two or more individuals who share management and profits.

Limited Liability

A legal structure that limits the financial responsibility of owners or shareholders to their investment in a company, protecting personal assets from business debts.

Carrying on Business

The act of engaging in commercial activities with a degree of continuity and organization, aiming at producing income or profit.