Evaluate the Summation sum from i=1 to 5 of 2i-3
The problem you are given asks you to calculate the total sum of a specific mathematical sequence. The sequence is defined by the expression 2i-3, where "i" represents each number in the range from 1 to 5. You are expected to find the value of this expression for each integer value of "i" within this range, and then add all those values together to obtain the final summation result. This task involves both substitution and addition as you work through the sequence.
$\sum_{i = 1}^{5} 2 i - 3$
Answer
Solution:
Step 1: Expansion of the Series
Write out the terms of the series by substituting each value of $i$ from 1 to 5 into the expression $2i - 3$. The series becomes:
$2 \cdot 1 - 3 + 2 \cdot 2 - 3 + 2 \cdot 3 - 3 + 2 \cdot 4 - 3 + 2 \cdot 5 - 3$
Step 2: Calculation of the Sum
Calculate the sum of the terms in the expanded series to get the final result:
$-3 - 3 - 3 - 3 - 3 + 2 \cdot (1 + 2 + 3 + 4 + 5) = -15 + 2 \cdot 15 = 15$
Knowledge Notes:
The problem at hand involves evaluating a finite arithmetic series. An arithmetic series is the sum of the terms of an arithmetic sequence, which is a sequence of numbers in which the difference between consecutive terms is constant. In this case, the sequence is generated by the expression $2i - 3$, where $i$ takes on the values from 1 to 5.
Key concepts to understand when solving this problem include:
Arithmetic Sequence: A sequence of numbers such that the difference of any two successive members is a constant. For example, in the sequence 2, 4, 6, 8, ..., the common difference is 2.
Arithmetic Series: The sum of the terms of an arithmetic sequence. It can be found by multiplying the average of the first and last term by the number of terms, or by summing each term individually as done in this problem.
Sigma Notation ($\Sigma$): A compact way to represent the summation of a series. The notation $\sum_{i=1}^{n} a_i$ represents the sum of $a_i$ from $i=1$ to $i=n$.
Substitution: In the context of series, substitution involves replacing the variable in the expression with each term in the sequence to find the individual terms of the series.
Simplification: The process of combining like terms and performing arithmetic operations to reduce an expression to its simplest form.
In this problem, the series is expanded by substituting values from 1 to 5 into the given expression, and then the terms are summed to find the total. The arithmetic properties of addition and multiplication are used to simplify the expression and arrive at the final result.
