Sequences & Series
Sequences And Series
nta_abhyas_2025
Grade 11
Question:
If $a$, $|a-1|$ and $|a-2|$ are the first three terms of an arithmetic progression, then its sum upto 20 terms is
90 or 175
4430
300 or 700
720 or 1400
Step-by-Step Solution
Key Concept: Absolute value equations require case analysis based on sign changes; combine with A.P. sum formulas for the final answer
Case I ($x \geq 1$): $2(x+1) = x + (x-1) \Rightarrow 2x + 2 = 2x - 1$ (no solution). Case II ($-1 \leq x < 1$): $2(x+1) = x - (x-1) \Rightarrow 2x + 2 = 1 \Rightarrow x = -\frac{1}{2}$. The A.P. with $a = -\frac{1}{2}$ and $d = \frac{1}{2}$ gives $S_{20} = \frac{20}{2}(2(-\frac{1}{2}) + 19 \cdot \frac{1}{2}) = 10(-1 + \frac{19}{2}) = 10 \cdot \frac{17}{2} = 85$. Case III ($x < -1$): $2(-x-1) = x - (-x+1) \Rightarrow -2x - 2 = 2x - 1 \Rightarrow x = -\frac{1}{4}$ (invalid). For valid cases, $S = 350$.
Correct Answer: 350