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------------------------------------------------ The sum of the 5th and the 9th terms of an arithmetic progression (AP) is 30. If its 25th term is three times its 8th term, find:

A) The common difference of the AP.

B) The first term of the AP.

Answer :

Final answer:

The common difference of the arithmetic progression is 30/13, and the first term is 30/26.

Explanation:

According to the problem statement, the sum of the 5th and 9th terms is 30, or a5 + a9 = 30.

Plugging in the formula for the nth term:

  • a5 = a1 + 4d
  • a9 = a1 + 8d

Adding these up gives us: 2a1 + 12d = 30 (Equation 1).

It is also given that the 25th term is three times the 8th term, or a25 = 3a8. Again, substituting the formula for the nth term gives us:

  • a25 = a1 + 24d
  • a8 = a1 + 7d

Setting a25 = 3a8 and solving for a1 and d, we get a1 + 24d = 3(a1 + 7d), which simplifies to a1 + 24d = 3a1 + 21d or 24d - 21d = 2a1 (Equation 2).

By solving Equation 1 and Equation 2 simultaneously:

  1. From Equation 2: d = 2a1
  2. Substitute d in Equation 1: 2a1 + 12(2a1) = 30 or 26a1 = 30, solving this we find a1 = 30/26
  3. Now, finding d, we get d = 2(30/26) or d = 60/26, reducing the fraction gives us d = 30/13

Subsequently, we find that the first term of the AP is 30/26 and the common difference is 30/13.

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