Problem 74 - Algebra and Trigonometry

Question

Find the sum of each sequence. \(\sum_{k=1}^{26}(3 k-7)\)

Step-by-Step Solution

Answer

The sum is 871.

Step 1: Understand the Problem

The problem requires finding the sum of the sequence \(\sum_{k=1}^{26}(3k-7)\). This is a summation problem with a specific formula given for the sequence terms.

Step 2: Split the Summation

Use the properties of summation to split the given sequence into two sums: \(\sum_{k=1}^{26}(3k) - \sum_{k=1}^{26}(7)\) . This makes the calculation more manageable.

Step 3: Calculate Individual Sums

Firstly, calculate \(\sum_{k=1}^{26} 3k\). This can be rewritten as 3 times the sum of the first 26 natural numbers: \(\sum_{k=1}^{26} k = \frac{26(26+1)}{2} = 351\)\. Thus, \(\sum_{k=1}^{26} 3k = 3 \times 351 = 1053\)\. Secondly, calculate \(\sum_{k=1}^{26} 7\). This is the sum of the constant term 7 repeated 26 times: \(\sum_{k=1}^{26} 7 = 7 \times 26 = 182\)\.

Step 4: Combine the Sums

Subtract the second sum from the first to get the final result: \(\sum_{k=1}^{26}(3k - 7) = 1053 - 182 = 871\).

Key Concepts

Sequence

A sequence is an ordered list of numbers that follow a specific pattern. In the given exercise, we deal with the sequence defined by the formula \(3k - 7\). This means that for each natural number value of \(k\), we get a term in the sequence by substituting it into the formula.

Let's illustrate how the sequence works:

• When \(k = 1\), the term is \(3 \times 1 - 7 = -4\).
• When \(k = 2\), the term is \(3 \times 2 - 7 = -1\).
• When \(k = 3\), the term is \(3 \times 3 - 7 = 2\).

This pattern continues until \(k = 26\). By following this pattern, the sequence becomes \(-4, -1, 2, 5, 8, ...\) and so on.

Series

A series is the sum of the terms of a sequence. In the problem, we need to find the sum of the sequence from \(k = 1\) to \(k = 26\). This is represented by the summation notation \(\sum_{k=1}^{26}(3k-7)\).

An example of a series could be adding the first few terms of the sequence we identified earlier:

• \(-4 + (-1) + 2 + 5 + ...\)

To simplify the calculation, we can split the series into two separate sums:

\(\sum_{k=1}^{26}(3k) - \sum_{k=1}^{26}7\). This is important as it allows us to handle simpler, more familiar series calculations.

Natural Numbers

Natural numbers are the set of positive integers starting from 1, 2, 3, and so on. They are counted rather than measured.

In the given problem, the natural numbers come into play when we calculate the sum of the first 26 natural numbers: \(\sum_{k=1}^{26} k\). We use the formula for the sum of the first \(n\) natural numbers, which is:

\(\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\).

In our case, \(n = 26\), making the sum: \(\sum_{k=1}^{26} k = \frac{26(26+1)}{2} = 351\). This sum is then multiplied by 3 based on the term formula from the sequence.

Summation Formula

A summation formula provides a method to quickly find the sum of a series without adding each term individually.

In this problem, we use two main summation formulas:

• \(\sum_{k=1}^{n} k\) for the sum of the first \(n\) natural numbers
• \(\sum_{k=1}^{n} c = nc\) for the sum of a constant repeated \(n\) times

Using these formulas, we find:

• \(\sum_{k=1}^{26} 3k = 3 \times 351 = 1053\)
• \(\sum_{k=1}^{26} 7 = 26 \times 7 = 182\)

Combining these results, we subtract the constant sum from the multiplied sum: \(1053 - 182 = 871\). So, the sum of the given sequence is \(871\).