Calculating the Sum of an Arithmetic Sequence with 24 Terms: 3, 9, and 15

Calculating the Sum of an Arithmetic Sequence with 24 Terms: 3, 9, and 15

One common task in mathematics is to find the sum of an arithmetic sequence. Given the first three terms of the sequence 3, 9, and 15, and knowing there are 24 terms in total, we can apply the steps and formulas to find the answer. This guide will walk you through the process of calculating the sum with the established arithmetic sequence formula.

Identifying the First Term and Common Difference

For an arithmetic sequence, the first term is denoted as (a), and the common difference is denoted as (d). Let's start by identifying these terms from the given sequence.

The first term (a) is 3. To determine the common difference (d), subtract the first term from the second term: d 9 - 3 6.

Finding the 24th Term

Next, we need to find the 24th term in the sequence using the formula for the (n)-th term of an arithmetic sequence:

[ a_n a (n - 1)d ]

Substituting the values, we find the 24th term as follows:

[ a_{24} 3 (24 - 1) times 6 3 23 times 6 3 138 141 ]

Calculating the Sum of the First 24 Terms

The sum of the first (n) terms of an arithmetic sequence is given by:

[ S_n frac{n}{2} times (a a_n) ]

Substituting (n 24), (a 3), and (a_{24} 141), we can calculate the sum:

[ S_{24} frac{24}{2} times (3 141) 12 times 144 1728 ]

Therefore, the sum of the arithmetic sequence with 24 terms is 1728.

Another Example: S26 with 3, 9, and 15

For another example, let's consider the same sequence but with a total of 26 terms. We apply the same process to solve for the sum:

The first term (a 3). The common difference (d 6). The number of terms (n 26).

Using the sum formula again:

[ S_{26} frac{26}{2} times [2 times 3 (26 - 1) times 6] ]

Substituting the values, we get:

[ S_{26} 13 times [6 150] 13 times 156 2028 ]

The sum of the 26 terms of the AP is thus 2028.

Verification and Summary

To summarize, the sum of an arithmetic sequence is calculated using the sum formula. The formulas and steps used here ensure accuracy and reliability. These examples clearly demonstrate the application of arithmetic sequence formulas in real-world problem-solving scenarios.