If the first term of an arithmetic sequence is 3 and the common difference is 2, find the sum of the first 10 terms.
Answer
The formula for the sum of the first n terms of an arithmetic sequence is \(S_n = \frac{n}{2}(a + a_n)\), where \(S_n\) is the sum, \(n\) is the number of terms, \(a\) is the first term, and \(a_n\) is the nth term. So the sum of the first 10 terms \(S_{10}\) is \(S_{10} = \frac{10}{2}(3 + 21) = 5(24) = 120\).
Step 1 ：The formula for the nth term of an arithmetic sequence is \(a_n = a + (n-1)d\), where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number. In this case, \(a = 3\) and \(d = 2\). So the 10th term \(a_{10}\) is \(a_{10} = 3 + (10-1)2 = 21\).
Step 2 ：The formula for the sum of the first n terms of an arithmetic sequence is \(S_n = \frac{n}{2}(a + a_n)\), where \(S_n\) is the sum, \(n\) is the number of terms, \(a\) is the first term, and \(a_n\) is the nth term. So the sum of the first 10 terms \(S_{10}\) is \(S_{10} = \frac{10}{2}(3 + 21) = 5(24) = 120\).
