Question

Find the sum of the first five terms of the AP $$1,5,9,13, \dots$$

Short answer

The sum of the first five terms of the given AP is 45.

Step-by-step solution

Identify the First Term

The first term in the arithmetic progression (AP) is given as 1. This is denoted by 'a'.

Determine the Common Difference

The common difference 'd' can be determined by subtracting the first term from the second term. For this AP, the second term is 5, so the common difference is 5 - 1 = 4.

Use the Formula for the Sum of an Arithmetic Sequence

The sum 'S_n' of the first n terms of an AP is given by the formula: \( S_n = \frac{n}{2} [2a + (n-1)d] \) where 'n' is the number of terms, 'a' is the first term, and 'd' is the common difference.

Calculate the Sum of the First Five Terms

Substitute 'n' with 5 (the number of terms), 'a' with 1 (the first term), and 'd' with 4 (the common difference) into the formula: \( S_5 = \frac{5}{2} [2 \cdot 1 + (5-1) \cdot 4] = \frac{5}{2} [2 + 16] = \frac{5}{2} \cdot 18 = 45. \)

Key concepts

Arithmetic Sequence

An arithmetic sequence, also known as an arithmetic progression, is a series of numbers in which each term after the first is obtained by adding a fixed, non-zero number called the common difference to the previous term. This definition is a foundation for many concepts in mathematics and is applied in various real-world scenarios like calculating time intervals or financial forecasting.

For example, in the sequence 1, 5, 9, 13... mentioned in the exercise, each number is four units apart from the next. This uniform interval between the terms is what defines it as an arithmetic sequence. Understanding this pattern allows students to anticipate subsequent terms, calculate specific terms within the sequence, and make logical deductions about the properties of the sequence.

Sum of an Arithmetic Sequence

The sum of an arithmetic sequence is the total when you add all the terms in the sequence. Calculating this sum involves a simple formula that uses the number of terms, the first term, and the common difference of the sequence. Knowing how to find the sum is useful not only for mathematical problem-solving but also for practical situations such as accounting for total expenses over a period if they rise at a steady rate.

The formula for the sum of the first n terms of an arithmetic sequence is:
\[ S_n = \frac{n}{2} [2a + (n-1)d] \]
where \(n\) is the number of terms, \(a\) is the first term, and \(d\) is the common difference. This formula is derived from the method of pairing terms equidistant from the start and end of the sequence, which always sum to a constant value.

Common Difference

The common difference in an arithmetic sequence is the fixed amount that each term increases by from the previous term. It is an essential characteristic that defines the sequence and can be found by subtracting the first term from the second term. The concept of the common difference is key to understanding and working with arithmetic progressions because it dictates the behavior of the sequence.

For instance, in the exercise provided, the common difference is 4, since each term after the first is greater by four than the term before it. This element of predictability allows for the common difference to be the anchor in using formulas that involve arithmetic sequences, such as those for finding a specific term or the sum of terms within the sequence. Recognizing and determining the common difference is a fundamental skill for algebra and arithmetic sequence problems.