Problem 30 - College Algebra

Question

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

Step-by-Step Solution

Answer

The sum of the series is 147.

Step 1: Understand the problem

The exercise is asking to calculate the sum of an arithmetic series. The series is formed by multiples of 7 (7i), where 'i' ranges from 1 to 6.

Step 2: Apply the formula of arithmetic series sum

The sum 'S' of an arithmetic series can be found by using the formula \( S = \frac{n}{2} [2a + (n-1)d ] \), where 'n' is the number of terms, 'a' is the first term, and 'd' is the difference between successive terms. Here, however, our series has terms that are multiples of '7', thus we modify the formula to \( S = 7 * \frac{n}{2} [2a + (n-1)d ] \)

Step 3: Substitute given values into the formula

Substitute the given information into the formula. Here, 'n' is 6 because it goes from 1 to 6, 'a' is 1, and 'd' is 1 because each subsequent term increases by 1. Thus, the formula becomes \( S = 7 * \frac{6}{2} [2(1) + (6-1)(1) ] \)

Step 4: Calculate the sum

Carry out the operations inside the brackets first, then multiply by 7. It results into \( S = 7 * 3 [ 2 + 5 ] = 7 * 3 * 7 = 147 \)