Find the sum of the first 50 terms of the arithmetic sequence: -10, -6, -2, 2,....

검증된 단계별 안내

Identify the first term \( a_1 \) of the arithmetic sequence. Here, the first term is \( -10 \).

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

Use the formula for the \( n \)-th term of an arithmetic sequence: \( a_n = a_1 + (n-1)d \). For the 50th term, write \( a_{50} = -10 + (50-1) \times 4 \).

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 \( n = 50 \), \( a_1 = -10 \), and the expression for \( a_{50} \) from the previous step.

Simplify the expression to find the sum \( S_{50} \) without calculating the final numeric value.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 constant, which helps identify the pattern and find any term.

Common Difference

The common difference is the fixed amount added to each term to get the next term in an arithmetic sequence. It is calculated 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 found 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 average of the first and last terms.