Find the sum of the first 20 terms of the arithmetic sequence: 4, 10, 16, 22,……….

Verified step by step guidance

1. Identify the first term \(a_1\) of the sequence, which is 4.
2. Determine the common difference \(d\) by subtracting the first term from the second term: \(d = 10 - 4 = 6\).
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 \(n = 20\), \(a_1 = 4\), and \(d = 6\) into the formula: \(S_{20} = \frac{20}{2} (2 \times 4 + (20-1) \times 6)\).
5. Simplify the expression inside the parentheses and then multiply by \(\frac{20}{2}\) to find the sum.

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. For example, in the sequence 4, 10, 16, 22, the common difference is 6, as each term increases by 6 from the previous term.

Formula for the 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, n is the number of terms, a_1 is the first term, and a_n is the nth term. This formula simplifies the process of finding the total sum without needing to add each term individually.

Finding the nth Term

The nth term of an arithmetic sequence can be found using the formula a_n = a_1 + (n - 1)d, where a_n is the nth term, a_1 is the first term, d is the common difference, and n is the term number. This allows us to determine the last term needed for the sum calculation in the sequence.