Find the number of terms in each series, the first term, and the last term. Then evaluate the sum. \sum _{n=1}^3\left(17n-25\right)
There are 3 terms in the series. The first term is found by plugging in n=1:
17(1) - 25 = -8
The last term is found by plugging in n=3:
17(3) - 25 = 26
To find the sum, we add up all three terms:
-8 + (17(2) - 25) + (17(3) - 25)
Simplifying:
-8 + 9 + 26 = 27
Therefore, the series has 3 terms, with a first term of -8, a last term of 26, and a sum of 27.
To find the number of terms in the series, we need to determine the range of values for the variable n. In this case, the series goes from n = 1 to n = 3.
Let's calculate the first term of the series by substituting n = 1 into the given expression:
\(17n - 25\) becomes \(17(1) - 25\) which gives us \(17 - 25 = -8\).
Now, let's calculate the last term by substituting n = 3:
\(17n - 25\) becomes \(17(3) - 25\) which gives us \(51 - 25 = 26\).
Therefore, the first term of the series is -8, and the last term is 26.
To find the number of terms in the series, we subtract the initial value from the final value and add 1: \(3 - 1 + 1 = 3\).
Hence, there are 3 terms in the series, and the first term is -8 and the last term is 26.
Now, let's evaluate the sum of the series.
The sum of the series \(\sum_{n=1}^{3}(17n-25)\) can be calculated using the formula \(S = \frac{n}{2}(a + l)\), where \(S\) is the sum, \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term.
Substituting the given values, we have:
\(S = \frac{3}{2}(-8 + 26)\)
Simplifying further:
\(S = \frac{3}{2}(18)\)
\(S = 27\)
Therefore, the sum of the series is 27.