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What is the 15th term from the last term of the arithmetical progression 2, 6, 10, ..., 86?

Published in Arithmetic Sequences 2 mins read

The 15th term from the last term of the arithmetic progression 2, 6, 10, ..., 86 is 30.

Here's how we arrive at this answer, drawing from the provided reference:


The reference states that the 15th term from the last in the AP 2, 6, 8, ... 86 is 30. We have a slight discrepancy in the provided sequence from the question (2, 6, 10, ..., 86) vs the reference (2, 6, 8, ..., 86). However, this difference in the third term does not affect the process or answer since the 15th term from the end is calculated regardless of that term. We can assume, therefore, the reference is correct and the sequence is actually 2, 6, 10, ..., 86. Let's analyze the arithmetical progression to clarify.


Understanding Arithmetic Progressions (APs)

An arithmetic progression (AP) is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference.

  • First Term (a): The first number in the sequence. In this case, a = 2.
  • Common Difference (d): The constant difference between consecutive terms. Here, d = 6 - 2 = 4.
  • Last Term (l): The final number in the sequence, which is 86.

Finding the nth Term from the End

To find the nth term from the end of an AP, we can use a simple technique:

  1. Reverse the AP: Treat the last term as the first term, and reverse the common difference.

  2. Calculate the new common difference: The new common difference will be the negative of the original common difference. So instead of +4, it's -4.

  3. Calculate the nth term: Use the standard formula for the nth term of an AP, where 'a' is the last term of the original AP, and 'd' is the reversed common difference:

    • nth term from end = l + (n - 1) * (-d)

Applying the Method to Our Problem

In our case:

  • l (last term) = 86
  • n (term from the end) = 15
  • d (common difference) = 4

Using the formula:

  • 15th term from the end = 86 + (15 - 1) * (-4)
  • 15th term from the end = 86 + 14 * (-4)
  • 15th term from the end = 86 - 56
  • 15th term from the end = 30

Therefore, the 15th term from the last of the AP 2, 6, 10, ... 86 is 30.


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