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How to Find the nth Term in a Sequence?

Published in Sequence Terms 2 mins read

Finding the nth term in a sequence involves identifying the pattern and expressing it as a formula. This formula allows you to calculate any term in the sequence without having to list out all the preceding terms.

Understanding Arithmetic Sequences

An arithmetic sequence is a series of numbers where the difference between consecutive terms remains constant. This constant difference is known as the common difference.

Steps to find the nth term in an arithmetic sequence:

  1. Identify the common difference (d): Find the difference between any two consecutive terms.

    • For example, in the sequence 2, 7, 12, 17... the common difference is 7 - 2 = 5.

  2. Determine the 'zeroth' term: This is the term that would come before the first term by subtracting the common difference from the first term.

    • Continuing with our sequence, the zeroth term would be 2 - 5 = -3.

  3. Use the formula: The general formula for the nth term (an) of an arithmetic sequence is:

    *an = a0 + (n d)**
    Where:

    • an is the nth term.
    • a0 is the zeroth term.
    • n is the term number (1 for the first term, 2 for the second term, etc).
    • d is the common difference.

Example:

Let's apply the formula to the sequence 2, 7, 12, 17...

  • The common difference (d) is 5.
  • The zeroth term (a0) is -3.
  • So, the nth term formula is: *an = -3 + (n 5)**

To find the 10th term (a10), substitute n = 10:

  • a10 = -3 + (10 * 5) = -3 + 50 = 47

Therefore, the 10th term in the sequence is 47.

Practical Insights:

  • The zeroth term is a conceptual tool to simplify the formula. It is the value before the initial term using the same common difference, as highlighted in the reference ![Part of a video titled Find the nth term in a sequence - YouTube]().
  • This method works only for arithmetic sequences with a constant difference.
  • It simplifies the calculation of any term, especially large ones, without listing out every single term.

Conclusion

By identifying the common difference, finding the zeroth term, and using the formula *an = a0 + (n d)**, you can effectively determine the nth term of any arithmetic sequence.

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