Evaluate the Summation sum from n=3 to 9 of n+2
The problem is asking for the evaluation of a specific summation. A summation is an operation that adds up a sequence of numbers. This particular summation requires you to calculate the sum of the terms 'n + 2' where 'n' takes on integer values starting from 3 and ending at 9. So, you would sum the results of that expression for each value of 'n' in that range: for n=3, then n=4, and so on, up to n=9. The question is looking for the final total of these calculations.
$\sum_{n = 3}^{9} n + 2$
Answer
Solution:
Step 1
Transform the original summation to start from $n=1$ by subtracting the unwanted terms.
$\sum_{n = 3}^{9} (n + 2) = \left( \sum_{n = 1}^{9} (n + 2) \right) - \left( \sum_{n = 1}^{2} (n + 2) \right)$
Step 2
Calculate the summation from $n=1$ to $n=9$ for the expression $n+2$.
Step 2.1
Decompose the summation into two separate summations.
$\sum_{n = 1}^{9} (n + 2) = \left( \sum_{n = 1}^{9} n \right) + \left( \sum_{n = 1}^{9} 2 \right)$
Step 2.2
Find the sum of the first nine natural numbers.
Step 2.2.1
Use the arithmetic series formula.
$\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$
Step 2.2.2
Plug in $n=9$ into the formula.
$\frac{9(9 + 1)}{2}$
Step 2.2.3
Perform the calculations.
Step 2.2.3.1
Combine $9$ and $1$.
$\frac{9 \cdot 10}{2}$
Step 2.2.3.2
Multiply $9$ by $10$.
$\frac{90}{2}$
Step 2.2.3.3
Divide $90$ by $2$.
$45$
Step 2.3
Sum the constant term $2$, repeated nine times.
Step 2.3.1
Apply the constant series formula.
$\sum_{k = 1}^{n} c = c \cdot n$
Step 2.3.2
Insert the values into the formula.
$(2)(9)$
Step 2.3.3
Multiply $2$ by $9$.
$18$
Step 2.4
Combine the results from the two summations.
$45 + 18$
Step 2.5
Add $45$ and $18$ together.
$63$
Step 3
Compute the summation from $n=1$ to $n=2$ for the expression $n+2$.
Step 3.1
Write out the terms of the series explicitly.
$1 + 2 + 2 + 2$
Step 3.2
Simplify the series.
Step 3.2.1
Add $1$ and $2$.
$3 + 2 + 2$
Step 3.2.2
Combine $2$ and $2$.
$3 + 4$
Step 3.2.3
Sum $3$ and $4$.
$7$
Step 4
Substitute the computed summations into the transformed summation.
$63 - 7$
Step 5
Subtract $7$ from $63$ to get the final result.
$56$
Knowledge Notes:
Summation notation ($\Sigma$) is used to denote the sum of a sequence of numbers.
The summation of a linear sequence, where each term is an integer $n$, can be calculated using the formula $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$.
The summation of a constant $c$ over $n$ terms is given by $\sum_{k = 1}^{n} c = c \cdot n$.
When evaluating a summation, you can split it into simpler summations that are easier to calculate, especially when dealing with polynomials or constants.
The arithmetic series formula is derived from the sum of the first $n$ natural numbers, which forms a linear sequence.
The constant series formula is straightforward since it involves adding the same number multiple times, which is equivalent to multiplication.
To evaluate a summation with a different starting index, you can adjust the summation to start from 1 and then subtract the terms that are not part of the original summation.
