Problem

Find the sum of the terms in the arithmetic sequence below, $S_{n}$.
\[
-3,-7,-11,-15, \ldots,-59
\]

Answer

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Answer

Final Answer: The sum of the terms in the arithmetic sequence is \(\boxed{-465}\).

Steps

Step 1 :First, we need to find the number of terms in the sequence. We can use the formula for the nth term of an arithmetic sequence, which is: \(a_{n} = a_{1} + (n - 1)d\), where \(a_{n}\) is the nth term, \(a_{1}\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms.

Step 2 :We can solve this equation for \(n\) to find the number of terms in the sequence. Given that the first term \(a_{1} = -3\), the last term \(a_{n} = -59\), and the common difference \(d = -4\), we can substitute these values into the formula to get \(n = (an - a1) / d + 1\), which gives us \(n = 15.0\).

Step 3 :Next, we can substitute the values into the formula for the sum of an arithmetic sequence to find the sum. The formula for the sum of an arithmetic sequence is \(S_{n} = n / 2 * (a_{1} + a_{n})\). Substituting the values we get \(S_{n} = -465.0\).

Step 4 :Final Answer: The sum of the terms in the arithmetic sequence is \(\boxed{-465}\).

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