Evaluate the Summation sum from i=1 to 5 of 3i-1
The question provided is asking you to calculate the sum of a sequence of numbers generated by a specific formula, 3i - 1, where "i" represents each integer from 1 to 5. In other words, you're being asked to substitute "i" with each of the integers in the given range (1 through 5) into the formula, calculate the value for each substitution, and then add all those values together to get the final sum.
$\sum_{i = 1}^{5} 3 i - 1$
Answer
Solution:
Step 1: Expansion of the Series
Write out the terms of the summation for each integer value of $i$ from 1 to 5.
$3 \cdot 1 - 1 + 3 \cdot 2 - 1 + 3 \cdot 3 - 1 + 3 \cdot 4 - 1 + 3 \cdot 5 - 1$
Step 2: Calculation of the Sum
Compute the sum of the terms obtained in the expansion.
$3(1) - 1 + 3(2) - 1 + 3(3) - 1 + 3(4) - 1 + 3(5) - 1 = 40$
Knowledge Notes:
The question involves evaluating a finite arithmetic series. An arithmetic series is the sum of terms in an arithmetic sequence, where each term after the first is formed by adding a constant to the previous term. The given series is a linear function of $i$, specifically $3i - 1$.
To solve the problem, we follow these steps:
Expansion of the Series: We expand the series by substituting the values of $i$ from 1 to 5 into the given function $3i - 1$. This gives us a series of terms that we can add together.
Simplification: We add the terms together to find the sum. Since the series is short, we can do this by simple arithmetic rather than using the formula for the sum of an arithmetic series.
The formula for the sum of an arithmetic series is not necessary in this case, but for reference, it is given by:
\[ S_n = \frac{n}{2}(a_1 + a_n) \] where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the nth term.
In this problem, the series is not strictly an arithmetic sequence because each term has a constant subtracted from it, but the approach to solving it is similar. We simply add up the terms after expanding them with the given values of $i$.
