Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for a_nto find a₂₀, the 20th term of the sequence. 1, 5, 9, 13,...

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Identify the first term of the arithmetic sequence, denoted as \(a_1\). In this sequence, the first term is \(1\).

Determine the common difference \(d\) by subtracting the first term from the second term: \(d = 5 - 1\).

Write the formula for the general term (the \(n\)th term) of an arithmetic sequence, which is \(a_n = a_1 + (n - 1) \times d\).

Substitute the values of \(a_1\) and \(d\) into the formula to get the explicit formula for this sequence.

Use the formula to find the 20th term by substituting \(n = 20\) into the general term formula.

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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. For example, in the sequence 1, 5, 9, 13,... the common difference is 4. Understanding this pattern is essential to formulating the general term.

General Term Formula of an Arithmetic Sequence

The general term (nth term) of an arithmetic sequence can be expressed as 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. This formula allows direct calculation of any term without listing all previous terms.

Evaluating the nth Term

Once the general term formula is established, substituting the desired term number (like n=20) into the formula gives the specific term's value. This step avoids recursion and efficiently finds terms deep in the sequence.

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for an to find a20, the 20th term of the sequence. 1, 5, 9, 13,...