Problem 30 - College Algebra

Question

Find each indicated sum. $$\sum_{i=1}^{6} 7 i$$

Step-by-Step Solution

Answer

The sum \(\sum_{i=1}^{6} 7i\) is 147.

Step 1: Identify the sum

We recognize this as the sum of the first 6 terms of the arithmetic sequence 7, 14, 21, ..., i.e., \(7 \sum_{i=1}^{6} i\). This is because each term is 7 times the term number.

Step 2: Apply the formula for the sum of the first n natural numbers

Now that the sum has been simplified, apply the formula \(n(n+1)/2\) for the sum of the first n natural numbers, where n = 6. This gives \[7 \times \frac{6(6+1)}{2}\].

Step 3: Calculate the final sum

By carrying out the computation in the expression from step 2, we find the sum. The result is \[7 \times \frac{6 \times 7}{2} = 7^2 \times 6/2 = 21 \times 7 = 147\].