Evaluate the Summation sum from k=2 to 6 of k^2+5
The question asks for an evaluation of a finite mathematical summation. Specifically, it's requesting the calculation of the total sum of the expression k^2+5 for each integer value of k starting at 2 and ending at 6. This means you need to substitute the values 2, 3, 4, 5, and 6 into the expression, calculate the sum for each, and then add all those sums together to get the final result. The expression involves an exponentiation operation and an addition operation for each term of the summation.
$\sum_{k = 2}^{6} k^{2} + 5$
Answer
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:
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.
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.
