Evaluate the Summation sum from i=1 to 15 of 2i
The problem is asking for the evaluation of a finite mathematical series. Specifically, you are asked to calculate the sum of a sequence of numbers generated by the formula 2i, where i represents each integer from 1 to 15. The series in question is an arithmetic sequence because the difference between consecutive terms is constant. The task is to find the total sum when you double each integer from 1 to 15 and then add all those doubled numbers together.
$\sum_{i = 1}^{15} 2 i$
Answer
Solution:
Step 1:
Extract the constant $2$ from the summation: $2 \sum_{i = 1}^{15} i$
Step 2:
Apply the arithmetic series sum formula: $\sum_{i = 1}^{n} i = \frac{n(n + 1)}{2}$
Step 3:
Insert the upper limit of the summation into the formula, multiplying by the constant extracted earlier: $2 \left( \frac{15(15 + 1)}{2} \right)$
Step 4:
Proceed with the simplification:
Step 4.1:
Begin simplifying the arithmetic inside the formula.
Step 4.1.1:
Add together $15$ and $1$: $2 \times \frac{15 \times 16}{2}$
Step 4.1.2:
Compute the product of $15$ and $16$: $2 \times \left( \frac{240}{2} \right)$
Step 4.2:
Eliminate the common factor of $2$.
Step 4.2.1:
Remove the common factor of $2$: $\cancel{2} \times \left( \frac{240}{\cancel{2}} \right)$
Step 4.2.2:
Finalize the expression: $240$
Knowledge Notes:
The problem involves evaluating a summation of a linear sequence. The key knowledge points involved in solving this problem are:
Summation Notation: The summation notation $\sum$ is used to denote the sum of a sequence of numbers. The expression $\sum_{i=1}^{n} a_i$ represents the sum of the sequence $a_i$ from $i=1$ to $i=n$.
Factoring Constants: In a summation, a constant multiplier can be factored out and placed in front of the summation symbol. For example, $\sum_{i=1}^{n} ca_i = c\sum_{i=1}^{n} a_i$.
Arithmetic Series: An arithmetic series is the sum of the terms of an arithmetic sequence, which is a sequence of numbers with a constant difference between consecutive terms. The sum of the first $n$ natural numbers is given by the formula $\sum_{i=1}^{n} i = \frac{n(n + 1)}{2}$.
Simplification: Simplification involves performing arithmetic operations such as addition, multiplication, and division to reduce expressions to a simpler form.
Cancellation: When the same factor appears in both the numerator and denominator, it can be canceled out to simplify the fraction.
By applying these concepts, we can evaluate the given summation efficiently.
