Data sufficiency – arithmetic progression: What is the first term of an arithmetic progression (A.P.) of positive integers? Statement I: The sum of the squares of the first and the second terms is 116. Statement II: The fifth term of the A.P. is divisible by 7. Decide which statement(s) provide sufficient data to determine the first term.

Introduction / Context:
This is a data sufficiency question involving an arithmetic progression (A.P.) whose terms are positive integers. Instead of directly solving for the first term, you must decide whether each given statement provides enough information to uniquely determine that first term. Data sufficiency questions test logical completeness rather than heavy computation.

Given Data / Assumptions:

We have an A.P. with first term a and common difference d, both integers, and all terms are positive.
Statement I: a^2 + (a + d)^2 = 116.
Statement II: The fifth term a + 4d is divisible by 7.
We assume standard properties of A.P. and that answers must be unique to be “sufficient”.

Concept / Approach:
For statement I, we treat it as a Diophantine (integer) equation and check if there is more than one valid pair (a, d) with positive terms. If there is exactly one valid pair, I alone is sufficient. For statement II, we observe that divisibility by 7 is a weak condition; many different A.P.s can satisfy it. We also check whether combining I and II changes anything about uniqueness.

Step-by-Step Solution:
From Statement I: Let the first term be a and the second term be a + d. We have a^2 + (a + d)^2 = 116. Look for integer pairs (x, y) such that x^2 + y^2 = 116 with x < y (first term smaller than second term). The squares less than 116 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The only pair of squares that sums to 116 is 16 + 100, corresponding to x = 4 and y = 10. So the first term a = 4, the second term a + d = 10, giving d = 6. All terms are positive integers, so this solution is valid and unique. Therefore, Statement I alone is sufficient to determine the first term. From Statement II: a + 4d is divisible by 7. There are infinitely many integer solutions (for example, (a, d) = (7, 0), (3, 1), (10, 2), etc., restricted only by positivity). Hence, Statement II alone is not sufficient. Combining I and II does not create a second solution; we already have a unique pair from I. So the combination does not change the conclusion that I alone is sufficient.

Verification / Alternative check:
After identifying a = 4 and d = 6 from Statement I, the fifth term is a + 4d = 4 + 24 = 28, which is divisible by 7, so Statement II is also true for this A.P. But the key point is that Statement I already pins down the first term uniquely. There is no other A.P. satisfying Statement I, so sufficiency is clear.

Why Other Options Are Wrong:
Saying only Statement II is sufficient is wrong because divisibility by 7 is too weak a condition. Claiming that both statements are necessary ignores the fact that Statement I alone already gives a unique solution. Saying that neither is sufficient overlooks the explicit unique solution provided by Statement I. There is no inconsistency between the statements, so the “data inconsistent” option is also incorrect.

Common Pitfalls:
A common error is to be intimidated by the equation and assume multiple solutions exist without checking. Another is to assume that whenever two statements are given, the exam expects you to combine them. In data sufficiency, always test each statement independently for uniqueness before considering combinations.

Final Answer:
Statement I alone determines the first term of the A.P. uniquely. The correct choice is Only statement I is sufficient.