Arithmetic-geometric Progression

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The sum of the series 1 + 2x + 3x² + 4x³ + ... to n terms (where |x| < 1) is

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1. **Identify Type**: This is an Arithmetico-Geometric Progression (AGP). 
2. **Derivation Technique**: Let $S = 1 + 2x + 3x^2 + \dots + nx^{n-1}$.
   - $xS = x + 2x^2 + \dots + (n-1)x^{n-1} + nx^n$.
3. **Subtract**: $S(1 - x) = 1 + x + x^2 + \dots + x^{n-1} - nx^n$.
4. **Apply GP Sum**: 
   - $S(1 - x) = \frac{1 - x^n}{1 - x} - nx^n$.
5. **Simplify**: 
   - $S = \frac{1 - x^n - nx^n(1 - x)}{(1 - x)^2} = \frac{1 - x^n - nx^n + nx^{n+1}}{(1 - x)^2}$
   - Result $= \frac{1 - (n+1)x^n + nx^{n+1}}{(1 - x)^2}$.

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Arithmetic-geometric Progression

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