In the sequence 21,700, 23,172, 24,644, 26,116,... which term is 314,628?

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Identify the type of sequence given. Since the difference between consecutive terms is constant, this is an arithmetic sequence.

Calculate the common difference $d$ by subtracting the first term from the second term: $d = 23,172 - 21,700$.

Use the formula for the $n$-th term of an arithmetic sequence: $a_n = a_1 + (n - 1) \times d$, where $a_1$ is the first term and $d$ is the common difference.

Set $a_n$ equal to 314,628 and substitute $a_1$ and $d$ into the formula: $314,628 = 21,700 + (n - 1) \times d$.

Solve the equation for $n$ to find which term in the sequence is 314,628.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. Recognizing this pattern helps in identifying the common difference and formulating the general term.

General Term Formula of an Arithmetic Sequence

The nth term of an arithmetic sequence can be expressed as a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference. This formula allows us to find any term in the sequence given its position.

Solving for the Term Number

To find the position n of a specific term value, substitute the term into the general formula and solve the resulting equation for n. This involves algebraic manipulation to isolate n and verify it is a positive integer.

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