Arithmetic-geometric Progression
The sum of the series 1 + 2x + 3x² + 4x³ + ... to n terms (where |x| < 1) is
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About This Question
- Subject
- mathematics
- Chapter
- sequence and series
- Topic
- arithmetic-geometric progression
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
[1 - (n+1)xⁿ + nxⁿ⁺¹]/(1-x)²
- Identify Type: This is an Arithmetico-Geometric Progression (AGP).
- Derivation Technique: Let S=1+2x+3x2+⋯+nxn−1.
- xS=x+2x2+⋯+(n−1)xn−1+nxn.
- Subtract: S(1−x)=1+x+x2+⋯+xn−1−nxn.
- Apply GP Sum:
- S(1−x)=1−x1−xn−nxn.
- Simplify:
- S=(1−x)21−xn−nxn(1−x)=(1−x)21−xn−nxn+nxn+1
- Result =(1−x)21−(n+1)xn+nxn+1.
This hard difficulty mathematics question is from the chapter sequence and series, covering the topic of arithmetic-geometric progression. It appeared in the 2025 exam.
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