Problem 37 - Algebra and Trigonometry

Question

Find the sum of the first 50 terms of the arithmetic sequence: \(-10,-6,-2,2, \dots\)

Step-by-Step Solution

Answer

The sum of the first 50 terms of the arithmetic sequence is 4400.

Step 1: Identify the first term and the common difference

The first term, \(a_1\), of the sequence is -10 and the common difference, \(d\), is found by subtracting -10 from -6, which is 4.

Step 2: Calculate the 50th term

Using the formula for the nth term of an arithmetic sequence, \(a_n = a_1 + (n - 1)d\), where \(n\) is the number of terms, here 50. We substitute these values into the formula to find\[ a_{50} = -10 + (50 - 1) * 4 = 186 \]

Step 3: Substitute into the sum formula

We will substitute \(n = 50\), \(a_1 = -10\), and \(a_{50} = 186\) into the sum formula \(S_n = \frac{n}{2} (a_1 + a_n)\) to get:\[ S_{50} = \frac{50}{2} (-10 + 186) = 25 * 176 \]