Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p>Let <i>S</i><sub>n</sub>, <i>S</i><sub>2n</sub>, <i>S</i><sub>3n</sub> are respectively the sums of first n, 2n, 3n terms of an arithmetic progression, then <i>S</i><sub>3n</sub> =</p>
<p>(A) \(2(S_{2n} - S_n)\)</p>
<p>(B) \(3(S_{2n} - S_n)\)</p>
<p>(C) \(3(S_{2n} - S_n)\)</p>
<p>(D) \(6(S_{2n} - S_n)\)</p>
Step-by-Step Solution
Key Concept: In an AP, the sum of terms in consecutive equal intervals form an AP themselves. Use this property to relate Sāā to differences of partial sums.
<p><strong>Solution:</strong> For an AP with first term a and common difference d:</p><p>\(S_n = \frac{n}{2}(2a + (n-1)d)\)</p><p>\(S_{2n} = \frac{2n}{2}(2a + (2n-1)d) = n(2a + (2n-1)d)\)</p><p>\(S_{3n} = \frac{3n}{2}(2a + (3n-1)d)\)</p><p>Computing \(S_{2n} - S_n\): This gives the sum of terms from position (n+1) to 2n.</p><p>Similarly, the sum from (2n+1) to 3n equals \(S_{3n} - S_{2n}\).</p><p>By properties of AP, \(S_{3n} - S_{2n} = S_{2n} - S_n\) when differences are equal.</p><p>Therefore: \(S_{3n} = S_{2n} + (S_{2n} - S_n) = 2S_{2n} - S_n = 3(S_{2n} - S_n)\)</p><p>ā“ Answer is C.</p>
Correct Answer: C