Question:medium

The number of terms in an arithmetic progression is 20. The sum of the four middle terms is (– 22). What are the values of the first term and common difference?

Statement 1: The sum of the first four terms is 74.
Statement 2: The difference between the first term and the common difference is 26.

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The question itself already gives one equation in a and d (from the middle-four-terms sum); each statement just needs to add a second independent equation.
Updated On: Jul 20, 2026
  • If the data in statement (1) alone is sufficient to answer the question, but the data in statement (2) alone is not sufficient.
  • If the data in statement (2) alone is sufficient to answer the question, but the data in statement (1) alone is not sufficient.
  • If the data in both the statements together are needed to answer the question.
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question.
  • If the data in neither statement (1) nor statement (2) is sufficient to answer the question, and more data is needed.
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The Correct Option is D

Solution and Explanation

Treat this purely as a two-unknowns, two-equations problem. With 20 terms in the AP, the question itself supplies equation one: the four middle terms (the 9th through 12th) sum to 4a + 38d = -22, which reduces to 2a + 19d = -11. This equation exists independent of any statement, so we always start with one relationship between a and d already in hand; we just need one more independent equation to solve the system.

Statement 1 supplies exactly that: the first four terms sum to 4a + 6d = 74, or 2a + 3d = 37. Two equations, two unknowns, solved by elimination: subtracting gives 16d = -48, so d = -3, then a = (37 - 3(-3))/2 = (37+9)/2 = 23. A clean unique pair, so this statement alone closes the problem.

Statement 2 supplies a different second equation: a - d = 26. Substituting a = 26 + d into the base equation, 2(26+d) + 19d = -11 gives 52 + 21d = -11, so 21d = -63 and d = -3, then a = 26 - 3 = 23. Again a unique, fully determined pair.

Because the base equation from the question stem is always available, and each statement independently supplies a second linear equation that yields one unique (a, d) pair, both statements work on their own - the answer is (d), either statement alone is sufficient.
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