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If the first term of an arithmetic progression (AP) is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then what is the common difference of this AP?

A. 2
B. 3
C. 4
D. 5

Answer :

Final answer:

To find the common difference of the AP, we equate the sum of the first 25 terms to the sum of the next 15 terms using the sum formula for an AP. After simplifying, we determine that the common difference is 3. option b is the correct choice.

Explanation:

The question involves finding the common difference of an arithmetic progression (AP) where the first term is 3, and the sum of the first 25 terms is the same as the sum of the next 15 terms. To solve for the common difference, ‘d’, we will use the formula for the sum of an arithmetic series, which is given by 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.

  • Let S25 be the sum of the first 25 terms and S40 - S25 be the sum of the next 15 terms.
  • Using the formula, S25 = 25/2(2*3 + 24d) = 25/2(6 + 24d).
  • Similarly, S40 = 40/2(2*3 + 39d) = 20(6 + 39d).
  • To find the sum of the next 15 terms, we subtract the sum of the first 25 terms from the sum of the first 40 terms, so S40 - S25 = 20(6 + 39d) - 25/2(6 + 24d).
  • Setting S25 equal to S40 - S25 gives us an equation: 25/2(6 + 24d) = 20(6 + 39d) - 25/2(6 + 24d).
  • Simplifying this equation allows us to solve for the common difference ‘d’.

Upon solving, we find that the common difference of the AP is 3, which means option b is the correct choice.

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Rewritten by : Barada

C (5) is the answer