Question

A series \( S_1 \) of five positive integers is such that the third term is half the first term, and the fifth term is 20 more than the first term. In series \( S_2 \), the \( n \)th term defined as the difference between the \( (n+1) \)th term and the \( n \)th term of series \( S_1 \), is an arithmetic progression with a common difference of 30.

• A. 50
• B. 60
• C. 70
• D. None of these

Hint:

Use arithmetic progression rules to calculate the differences between terms in a sequence when dealing with series-related problems.

Answer 1

The Correct Option is B

Let the first term of \( S_1 \) be \( a \). The third term is half of the first term, so the third term is \( \frac{a}{2} \). The fifth term is 20 more than the first term, so the fifth term is \( a + 20 \). Thus, we have the following equations: \[ S_1 = \{a, \, b, \, \frac{a}{2}, \, d, \, a + 20\} \] The difference between successive terms in \( S_2 \) is an arithmetic progression with a common difference of 30. This means that the difference between the second term and the first term is 30. \[ S_2 = \left\{ \frac{a}{2} - a, \, a + 20 - \frac{a}{2} \right\} \] Solving for \( S_2 \), we get: \[ \frac{a}{2} - a = -\frac{a}{2}, \quad a + 20 - \frac{a}{2} = \frac{2a + 40 - a}{2} = \frac{a + 40}{2} \] Since \( S_2 \) is an arithmetic progression with a common difference of 30, the common difference can be calculated. From here, we get the second term of \( S_2 \) as 60. Hence, the second term of \( S_2 \) is \( \boxed{60} \). Thus, the answer is: b. 60.