Evaluate the Summation sum from i=1 to 5 of 5i-2
The question is asking for the calculation of a finite mathematical series. Specifically, it's requesting the computation of the accumulated sum of the expression "5i - 2" as the variable "i" ranges from 1 to 5. In other words, one must calculate the value of the expression for each integer value of "i" within that range, and then add all those results together to get the final sum.
$\sum_{i = 1}^{5} 5 i - 2$
Answer
Solution:
Step 1: Expand the Summation
Write out each term of the summation by substituting the values of $i$ from 1 to 5 into the expression $5i - 2$.
$$(5 \times 1 - 2) + (5 \times 2 - 2) + (5 \times 3 - 2) + (5 \times 4 - 2) + (5 \times 5 - 2)$$
Step 2: Calculate the Sum
Add the values of each term to find the total sum.
$$3 + 8 + 13 + 18 + 23 = 65$$
Knowledge Notes:
The problem 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 with a common difference between consecutive terms. In this case, the common difference is the coefficient of $i$ (which is 5), and the series is finite because it sums terms from $i=1$ to $i=5$.
The general form of an arithmetic series can be written as:
$$\sum_{i=1}^{n}(a + (i - 1)d)$$ where $a$ is the first term, $d$ is the common difference, and $n$ is the number of terms.
In the given problem, the expression $5i - 2$ represents an arithmetic sequence with a common difference of 5 and the first term being $5 \times 1 - 2 = 3$. The series is the sum of the first five terms of this sequence.
To solve the problem, we follow these steps:
Expand the series by computing each term using the given expression.
Simplify the expanded form by performing the addition to find the sum of the series.
In this case, the series is simple enough to be expanded and calculated directly without using the formula for the sum of an arithmetic series, which is:
$$S_n = \frac{n}{2}(a_1 + a_n)$$ where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the nth term. However, for longer series or when a pattern is not easily discernible, this formula can be very useful.
