High School

Find the first term of an arithmetic progression (AP) and the common difference in the series, given that the sum of the first 6 terms of the AP is 18. If the sum of the first 22 terms of the AP is 6, find the common difference.

Answer :

Final answer:

To find the first term and common difference of the AP, two equations are derived from the sums of the first 6 and 22 terms. By simplifying and solving these equations simultaneously, the first term and common difference can be calculated.

Explanation:

To find the first term of an arithmetic progression (AP) and the common difference, we must utilize the formula for the sum of the first n terms of an AP, which is Sn = n/2 (2a + (n - 1)d), where Sn is the sum of the first n terms, a is the first term, and d is the common difference.

Given that the sum of the first 6 terms (S6) is 18, and the sum of the first 22 terms (S22) is 6, we can set up two equations:

  1. 18 = 6/2 (2a + 5d)
  2. 6 = 22/2 (2a + 21d)

Solving these two equations simultaneously, we can find the values for a and d.

Step 1: Simplify the equations:

  • 18 = 3 (2a + 5d)
  • 6 = 11 (2a + 21d)

Step 2: Divide both equations by their coefficients to isolate the terms with a and d:

  • 6 = 2a + 5d
  • 6/11 = 2a + 21d

Step 3: Multiply both sides of the second equation by 11 to make the coefficients of the a terms equal:

  • 6 = 2a + 5d
  • 6 = 22a + 231d

By subtracting the second equation from the first, we find the common difference in the series.