Question

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.

• A. If the data in statement (1) alone is sufficient to answer the question, but the data in statement (2) alone is not sufficient.
• B. If the data in statement (2) alone is sufficient to answer the question, but the data in statement (1) alone is not sufficient.
• C. If the data in both the statements together are needed to answer the question.
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question.
• E. If the data in neither statement (1) nor statement (2) is sufficient to answer the question, and more data is needed.

Hint:

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.

Answer 1

The Correct Option is D

Let the first term be a and the common difference be d, so the 20 terms are a, a+d, a+2d, ... , a+19d. With 20 terms, the four middle terms are the 9th, 10th, 11th and 12th terms: (a+8d), (a+9d), (a+10d), (a+11d). Their sum is 4a + 38d, which the question tells us equals -22. Simplifying, this base equation is 2a + 19d = -11, and it is already known from the question itself, before looking at either statement.

Statement 1 gives the sum of the first four terms, a + (a+d) + (a+2d) + (a+3d) = 4a + 6d = 74, i.e. 2a + 3d = 37. Combined with the base equation 2a + 19d = -11, subtracting gives 16d = -48, so d = -3, and substituting back gives a = 23. This is a unique solution, so statement 1 alone (using the equation already given in the question) is sufficient.

Statement 2 gives a - d = 26. Combined with the base equation, substitute a = d + 26 into 2a + 19d = -11: 2(d+26) + 19d = -11, so 21d = -63, d = -3, and a = 23. Again a unique solution, so statement 2 alone is also sufficient.

Since each statement, combined with the base fact already stated in the question, independently pins down a unique pair (a, d), and both approaches even land on the same actual values (a = 23, d = -3), either statement alone is sufficient. The answer is (d).