How can I determine the nth term of an arithmetic sequence?

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Finding the nth term of an arithmetic sequence is a straightforward process once you understand the basic formula and components involved. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is referred to as the common difference, denoted by d.

The general formula for the nth term of an arithmetic sequence can be expressed as:

T_n = a + (n - 1) * d

In this formula:

  • T_n represents the nth term you want to find.
  • a is the first term of the sequence.
  • n is the position of the term in the sequence you wish to find.
  • d is the common difference between consecutive terms.

To illustrate how to use this formula, let’s go through an example:

Example:

Consider the arithmetic sequence: 3, 7, 11, 15, …

  • The first term a is 3.
  • The common difference d can be calculated by subtracting the first term from the second term: 7 – 3 = 4.

Now, if you wish to find the 10th term of this sequence (where n = 10), you would plug these values into the formula:

T_10 = 3 + (10 - 1) * 4

This simplifies to:

T_10 = 3 + 9 * 4

Further simplifying gives:

T_10 = 3 + 36 = 39

Thus, the 10th term of the sequence is 39.

In summary, to find the nth term of an arithmetic sequence, identify the first term and the common difference, and then apply the formula. This method can be used for any term in the sequence, making it a valuable technique in understanding arithmetic progressions!

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