Question

What is the first term of an arithmetic progression of positive integers?
Statement I
I. Sum of the squares of the first and the second term is 116.
Statement II
II. The fifth term is divisible by 7.

• A. The question can be answered with the help of statement I alone.
• B. The question can be answered with the help of statement II alone.
• C. Both statement I and statement II are needed to answer the question.
• D. The question cannot be answered even with the help of both the statements.

Hint:

When two unknowns appear in AP problems, one statement usually gives a numeric equation and the other gives a modular or divisibility condition.

Answer 1

The Correct Option is C

Step 1: Define the AP terms Let the first term be $a$ and common difference be $d$. Then first term = $a$, second term = $a + d$. Step 2: From Statement I $a^2 + (a + d)^2 = 116 \Rightarrow a^2 + a^2 + 2ad + d^2 = 116 \Rightarrow 2a^2 + 2ad + d^2 = 116$. This is one equation with two unknowns ($a, d$) → infinite solutions possible. Not sufficient. Step 3: From Statement II Fifth term = $a + 4d$ divisible by 7. Alone, this does not give unique $a$. Not sufficient. Step 4: Combining Statements From I: $2a^2 + 2ad + d^2 = 116$ and from II: $a + 4d = 7k$, $k$ integer. Substitute and solve simultaneously to get a unique $a$ (positive integer). Step 5: Conclusion Both statements are require(d)