Find the sum of the first 22 terms of the arithmetic sequence: 5, 12, 19, 26, ...

Verified step by step guidance

1. Identify the first term $a_1$ of the sequence, which is 5.
2. Determine the common difference $d$ by subtracting the first term from the second term: $d=12-5=7$.
3. Use the formula for the sum of the first $n$ terms of an arithmetic sequence: $S_n=\frac{n}{2}(2a_1+(n-1)d)$.
4. Substitute $n=22$, $a_1=5$, and $d=7$ into the formula: $S_{22}=\frac{22}{2}(2\times 5+(22-1)\times 7)$.
5. Simplify the expression inside the parentheses and then multiply by $\frac{22}{2}$ to find the sum of the first 22 terms.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. In the given sequence, the common difference can be calculated by subtracting any term from the subsequent term, which is 7 in this case (12 - 5 = 7). Understanding this concept is crucial for identifying the pattern and calculating further terms.

Formula for the n-th Term

The n-th term of an arithmetic sequence can be found using the formula: a_n = a_1 + (n - 1)d, where a_1 is the first term, d is the common difference, and n is the term number. For the sequence provided, the first term is 5 and the common difference is 7. This formula allows us to determine any term in the sequence, which is essential for finding the sum of the first 22 terms.

Sum of an Arithmetic Sequence

The sum of the first n terms of an arithmetic sequence can be calculated using the formula: S_n = n/2 * (a_1 + a_n), where S_n is the sum of the first n terms, a_1 is the first term, and a_n is the n-th term. Alternatively, it can also be expressed as S_n = n/2 * (2a_1 + (n - 1)d). This concept is vital for solving the question, as it provides a systematic way to compute the total of the first 22 terms.