To find the sum of the series
[tex]\[
\sum_{n=1}^{120}(3n - 7),
\][/tex]
we break it down into simpler parts using summation properties. Let's summarize the steps to solve it.
1. Identify the series: We need to find the sum of [tex]\(3n - 7\)[/tex] from [tex]\(n = 1\)[/tex] to [tex]\(n = 120\)[/tex].
2. Break the series into two separate sums:
[tex]\[
\sum_{n=1}^{120}(3n - 7) = \sum_{n=1}^{120} 3n - \sum_{n=1}^{120} 7
\][/tex]
3. Calculate the first sum [tex]\(\sum_{n=1}^{120} 3n\)[/tex]:
- Factor out the constant [tex]\(3\)[/tex]:
[tex]\[
\sum_{n=1}^{120} 3n = 3 \sum_{n=1}^{120} n
\][/tex]
- The sum of the first [tex]\(n\)[/tex] natural numbers is given by the formula [tex]\(\frac{n(n+1)}{2}\)[/tex].
[tex]\[
\sum_{n=1}^{120} n = \frac{120 \cdot 121}{2} = 7260
\][/tex]
- Multiply by [tex]\(3\)[/tex]:
[tex]\[
3 \sum_{n=1}^{120} n = 3 \cdot 7260 = 21780
\][/tex]
4. Calculate the second sum [tex]\(\sum_{n=1}^{120} 7\)[/tex]:
- Factor out the constant [tex]\(7\)[/tex]:
[tex]\[
\sum_{n=1}^{120} 7 = 7 \cdot 120 = 840
\][/tex]
5. Combine the results:
[tex]\[
\sum_{n=1}^{120} (3n - 7) = 21780 - 840 = 20940
\][/tex]
So, the sum
[tex]\[
\sum_{n=1}^{120}(3n - 7)
\][/tex]
is
[tex]\[
\boxed{20940}.
\][/tex]
