Answer
$\sum\limits_{i=1}^{11} 1.4n + 4.6 = 143$
Work Step by Step
To find the sum of a finite series, we can use the formula:
$S_n = \frac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum of the series, $n$ is the number of terms in the series, $a_1$ is the first term, and $a_n$ is the $nth$ term.
We already have $a_1$ and $n$; however, we need to find $a_n$. This can be done by using the explicit formula for arithmetic sequences, which is given by $a_n = a + (n - 1)d$. We can find $d$ by subtracting one term from the following term:
$d = 7.4 - 6 = 1.4$
Plug values into the explicit formula for an arithmetic sequence to find the explicit formula of this series:
$a_n = 6 + (n - 1)(1.4)$
Use distributive property:
$a_n = 6 + 1.4n - 1.4$
Combine like terms:
$a_n = 1.4n + 4.6$
Now, we can plug our values into the summation notation, where we plug in the explicit formula for finding the $nth$ term with the lower limit being $1$ and the upper limit being the value of $n$ for the last term in the series:
$\sum\limits_{i=1}^{11} 1.4n + 4.6$
To find the final term in the series, use the explicit formula for a finite arithmetic sequence:
$a_{11} = 6 + (11 - 1)(1.4)$
Evaluate expressions within parentheses first:
$a_{11} = 6 + (10)(1.4)$
Multiply:
$a_{11} = 6 + (14)$
Add to solve:
$a_{11} = 20$
Now we can plug the values that are known into the formula to find the sum of a finite arithmetic series:
$S_{11} = \frac{11}{2}(6 + 20)$
Evaluate what is in parentheses first:
$S_{11} = \frac{11}{2}(26)$
Multiply:
$S_{11} = \frac{286}{2}$
Simplify the fraction:
$S_{11} = 143$
