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

Verified step by step guidance

1. Identify the pattern in the sequence by finding the difference between consecutive terms.
2. Calculate the difference: 23,172 - 21,700 = 1,472.
3. Verify the difference is consistent: 24,644 - 23,172 = 1,472 and 26,116 - 24,644 = 1,472.
4. Recognize the sequence is arithmetic with a common difference of 1,472.
5. Use the formula for the nth term of an arithmetic sequence: a_n = a_1 + (n-1) * d, where a_1 is the first term and d is the common difference, to find n when a_n = 314,628.

Key Concepts

Arithmetic Sequences

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. Understanding this concept is crucial for identifying patterns in the sequence provided and for determining the general formula for the nth term.

Finding the nth Term

To find a specific term in a sequence, one can use the formula for the nth term of an arithmetic sequence, which is given by 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 us to calculate any term in the sequence based on its position.

Solving Equations

Once the formula for the nth term is established, solving for n when given a specific term (in this case, 314,628) involves setting the nth term formula equal to that value and rearranging the equation. This process requires algebraic manipulation skills to isolate n and find its value, which indicates the position of the term in the sequence.