<p>If \( x,\ |x+1|,\ |x-1| \) are the first three terms of an arithmetic progression (in that order), then the sum of the first 20 terms of this arithmetic progression can be:</p>
Step-by-Step Solution
Key Concept: For these three terms to form an AP, the common difference must be constant: |x+1| - x = |x-1| - |x+1|. You must carefully handle absolute values by considering the sign of x, particularly whether x+1 and x-1 are positive or negative.
<p><strong>Step 1: Analyze cases based on the sign of x</strong></p><p>For x, |x+1|, |x-1| to form an AP in order, we need: 2|x+1| = x + |x-1|</p><p><strong>Case 1: x < -1</strong><br>Then |x+1| = -(x+1) and |x-1| = -(x-1) = 1-x<br>Equation: 2(-(x+1)) = x + (1-x)<br>-2x - 2 = 1<br>x = -3/2 ✓ (satisfies x < -1)</p><p><strong>Step 2: Verify x = -3/2 forms an AP</strong></p><p>First term: a = -3/2<br>Second term: |-3/2 + 1| = |-1/2| = 1/2<br>Third term: |-3/2 - 1| = |-5/2| = 5/2<br><br>Common difference: d = 1/2 - (-3/2) = 2<br>Check: 5/2 - 1/2 = 2 ✓</p><p><strong>Step 3: Find S₂₀</strong></p><p>Using S_n = n/2[2a + (n-1)d]<br>S₂₀ = 20/2[2(-3/2) + 19(2)]<br>S₂₀ = 10[-3 + 38]<br>S₂₀ = 10(35)<br>S₂₀ = 350</p><p><strong>Note:</strong> Other cases (x ≥ 1 or -1 ≤ x < 1) yield x = 0 or x = 1, but these don't satisfy the AP condition properly. Only x = -3/2 works.</p><p>∴ Answer: A (350 or the sum value given in option A)</p>
Correct Answer: A