Question

If the sum of the first 11 terms of an arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms?

• A. 0
• B. -1
• C. 1
• D. Not unique

Hint:

For such problems, check if the sum equations lead to unique solutions or if multiple progressions are possible.

Answer 1

The Correct Option is D

The sum of an arithmetic progression \(S_n\) is given by: \[ S_n = \frac{n}{2} \left(2a + (n - 1) d \right) \] Let the first term be \(a\) and the common difference be \(d\). Given that the sum of the first 11 terms is equal to the sum of the first 19 terms, we have: \[ S_{11} = S_{19} \] This gives an equation that we can solve for \(a\) and \(d\). However, the solution is not unique because there are infinitely many arithmetic progressions that satisfy this equation. Thus, the sum of the first 30 terms is not uniquely determined. \[ \boxed{\text{Not unique}} \]