Problem 36
Question
Find the sum of the first 25 terms of the arithmetic sequence: \(7,19,31,43, \dots\)
Step-by-Step Solution
Verified Answer
The sum of the first 25 terms of the arithmetic sequence is 3775.
1Step 1: Identify the common difference, first term and the number of terms
From the sequence, the first term \(a_1 = 7\), the common difference \(d = 19 - 7 = 12\), and the number of terms \(n = 25\). We can see that the sequence is arithmetic because the difference between any two successive terms is constant.
2Step 2: Find the 25th term
Use the formula \(a_n = a_1 + (n - 1) * d\) to calculate the 25th term \(a_n\). So, \(a_n = 7 + (25 - 1) * 12 = 295\). Thus, the 25th term of the sequence is 295.
3Step 3: Calculate the sum of the first 25 terms
Now, use the sum formula for an arithmetic sequence: \(S_n = n/2 * (a_1 + a_n)\). Insert the values and calculate the sum: \(S_{25} = 25/2 * (7 + 295) = 25/2 * 302 = 3775\). Therefore, the sum of the first 25 terms is 3775.
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Problem 36
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