Problem 62 - Beginning and Intermediate Algebra

Question

Evaluate each sum using a formula for \(S_{n}\). $$\sum_{i=1}^{6}(3 i+8)$$

Step-by-Step Solution

Answer

The sum of the arithmetic series \(\sum_{i=1}^{6}(3i + 8)\) is 111.

Step 1: Identify the given series

First, let's identify the given sum: \(\sum_{i=1}^{6}(3i + 8)\) Here, the lower limit is 1 and the upper limit is 6. We can rewrite this sum as a series: \(3(1) + 8 + 3(2) + 8 + 3(3) + 8 + ... + 3(6) + 8\)

Step 2: Recognize that the given series is an arithmetic series

Next, look at the differences between consecutive terms. The common difference in the series is the coefficient of 'i', which is 3 in this case. Thus, the given series is an arithmetic series.

Step 3: Find the first term (a) and the last term (l) of the arithmetic series

To find the first term, substitute i=1 in the formula (3i + 8): a = 3(1) + 8 = 11 To find the last term, substitute i=6 in the formula (3i + 8): l = 3(6) + 8 = 26

Step 4: Calculate the sum using the arithmetic sum formula

To find the sum of an arithmetic series, we can use the formula: \(S_n = \dfrac{n}{2} (a + l)\) Where 'S_n' is the sum, 'n' is the number of terms, 'a' is the first term, and 'l' is the last term. In our case, n=6, a=11, and l=26. Substitute the values into the formula and evaluate: \(S_6 = \dfrac{6}{2} (11 + 26) = 3(37) = 111\) The sum of the given arithmetic series is 111.

Key Concepts

sum of series

In mathematics, summing up a sequence of numbers can sometimes be daunting. However, understanding the principles behind the sum of a series makes it manageable. Here, we explore an arithmetic series, which is a sequence where the difference between every consecutive number is constant.
For arithmetic series, the sum of the series can be efficiently calculated using specific formulas. This is especially handy when dealing with long sequences. The series we looked at was:

• The sum notation: \( \sum_{i=1}^{6}(3i + 8) \).
• Special cases like these can be broken down and simplified with arithmetic principles and using particular formulas.

Calculating the sum directly term by term would be laborious and prone to human error. Hence, using a structured approach and formulas is both time-effective and accurate.

arithmetic series formula

Arithmetic series have a special formula that brings simplicity to their evaluation. When you have a series where each term increases by the same constant value, use the arithmetic series formula to find the sum. The formula is:

• \( S_n = \frac{n}{2} (a + l) \).

Here:

• \( S_n \) represents the sum of the series.
• \( n \) is the number of terms.
• \( a \) is the first term.
• \( l \) is the last term.

By substituting these values, you can swiftly calculate the series' sum. For instance, in our example where \( n=6 \), \( a=11 \), and \( l=26 \), it was straightforward to compute the sum using:

• \( S_6 = \frac{6}{2}(11+26) = 3(37) = 111 \).

This step demystifies the calculation, empowering students to tackle such problems independently.

common difference

The common difference in an arithmetic series is pivotal for identifying its pattern. It is the fixed amount each term increases from the previous one. In essence, it constitutes the backbone of the arithmetic structure. For the series \( \sum_{i=1}^{6}(3i + 8) \), consider each term:

• The series expands to \( 3(1) + 8, 3(2) + 8, ..., 3(6) + 8 \).
• Here, the common difference is 3, evident from the expression \( 3i + 8 \).

Recognizing the common difference allows students to understand and predict subsequent terms easily. Moreover, it simplifies the calculation of sums using formulas, guiding you to derive other components like the last term \( l \).
This recognition is fundamental and a prelude to simplifying and evaluating the entire series.

evaluating series

Evaluating a series involves several systematic steps to determine the sum quickly and accurately. First, verify if the series is arithmetic by checking for a consistent common difference. Once confirmed,

• Identify the first term \( a \) and the last term \( l \).
• Use the arithmetic series formula \( S_n = \frac{n}{2} (a + l) \).

These initial steps organize the problem, setting the stage for applying the formula. For the example \( \sum_{i=1}^{6}(3i + 8) \), we saw:

• First term \( a = 11 \), last term \( l = 26 \), and six terms overall.

Substituting these into the formula, you calculated the sum efficiently to reach the answer \( S_6 = 111 \). Thus, evaluating series with these methods streamlines even complex calculations.