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

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Identify the first term \( a_1 \) of the arithmetic sequence. Here, \( a_1 = 4 \).

Determine the common difference \( d \) by subtracting the first term from the second term: \( d = 10 - 4 = 6 \).

Use the formula for the \( n \)-th term of an arithmetic sequence: \( a_n = a_1 + (n - 1)d \). For the 20th term, write \( a_{20} = 4 + (20 - 1) \times 6 \).

Apply the formula for the sum of the first \( n \) terms of an arithmetic sequence: \( S_n = \frac{n}{2} (a_1 + a_n) \). Substitute \( n = 20 \), \( a_1 = 4 \), and the expression for \( a_{20} \) from the previous step.

Simplify the expression to find the sum \( S_{20} \) without calculating the final numeric value.

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주요 개념

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Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant difference to the previous term. In this sequence, the difference between consecutive terms is fixed, which helps in identifying the pattern and calculating any term.

Common Difference

The common difference is the constant amount added to each term to get the next term in an arithmetic sequence. It is found by subtracting any term from the following term, and it is essential for determining the nth term and the sum of 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 * (first term + last term). This formula simplifies adding many terms by using the number of terms and the values of the first and last terms.