Evaluate the Summation sum from k=2 to 6 of k^2+5

Solution:

Step 1: Expansion of the Series

List out each term of the series by substituting the values of $k$ from 2 to 6 into the expression $k^2+5$.

$2^2+5,3^2+5,4^2+5,5^2+5,6^2+5$

Step 2: Calculation of the Sum

Add all the terms together to find the total sum.

$4+5+9+5+16+5+25+5+36+5=115$

Knowledge Notes:

The summation notation $\sum$ is a concise way to represent the addition of a sequence of numbers. The expression $\sum _{k=m}^{n}f(k)$ denotes the sum of $f(k)$ for all integers $k$ from $m$ to $n$, inclusive.

To evaluate a summation:

1. Expansion: Replace the summation index $k$ with each integer from the lower limit to the upper limit and write out each term of the sequence.

2. Simplification: Perform the necessary arithmetic operations to simplify and combine the terms.

In this problem, the summation is $\sum _{k=2}^6(k^2+5)$. The process involves two main steps:

• Expansion: We substitute $k$ with each integer from 2 to 6 into the expression $k^2+5$.

• Simplification: We then add all the resulting terms to find the total sum.

The expression $k^2+5$ represents a quadratic function where $k^2$ is the square of the integer $k$, and 5 is a constant added to each term. The series expansion lists all terms from $k=2$ to $k=6$, and the simplification step involves adding these terms to get the final result.