Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p>Let \(a_1, a_2, a_3, \ldots, a_n\) are in A.P., then</p>
<p>(A) \(a_1 - 2a_2 + a_3 = 0\)</p>
<p>(B) \(a_1 - 2a_2 + 2a_3 - 2a_4 + a_5 = 0\)</p>
<p>(C) \(a_1 + 3a_2 - 3a_3 - a_4 = 0\)</p>
<p>(D) \(a_1 - 4a_2 + 6a_3 - 4a_4 + a_6 = 0\)</p>
Step-by-Step Solution
Key Concept: In an A.P., the common difference relationship leads to a second-order finite difference being zero.
<p>If \(a_1, a_2, a_3, \ldots, a_n\) are in A.P., then \(a_2 - a_1 = a_3 - a_2\), which gives \(2a_2 = a_1 + a_3\). Therefore, \(a_1 - 2a_2 + a_3 = 0\).</p>
Correct Answer: A