Problem 49 - College Algebra

Question

Write out the first three terms and the last term. Then use the formula for the sum of the first \(n\) terms of an arithmetic sequence to find the indicated sum. $$\sum_{i=1}^{100} 4 i$$

Step-by-Step Solution

Answer

The sum of the first 100 terms of the sequence is 20200.

Step 1: Identify the first three terms and the last term

The given sequence is formed by multiplying the term number by 4. So, the first term is \(4*1 = 4\), the second term is \(4*2 = 8\), and the third term is \(4*3 = 12\). Now, to find the last term, multiply the term number 100 by 4, that gives \(4*100 = 400\).

Step 2: Use the arithmetic sequence sum formula

With the first term (\(a = 4\)) and the last term (\(l = 400\)), you can use the formula for the sum of an arithmetic sequence. This formula is \( \frac{n}{2} (a + l) \), where \(n\) is the number of terms and in this case, \(n = 100\) terms.

Step 3: Perform the calculation

Insert the identified values into the formula to find the sum \( S = \frac{100}{2} (4 + 400) = 20200 \), so the sum of the first 100 terms of the sequence is 20200.