Discussion

Explanation:

Let S = 1 + 2x + 3x² + 4x³ + ... + nx^n-1    ...(1)

This is an Arithmetic Geometric Progression (AGP). The coefficients are in AP where as the variable is in GP.

Step 1: Multiply the whole equation with x.
⇒ xS = x + 2x² + 3x³ + 4x⁴ + ... + nxⁿ    ...(2)

Now, (1) - (2)
⇒ (1 - x)S = 1 + (2x - x) + (3x² - 2x²) + (4x³ - 3x³) + ... + ((nx^n-1 - (n - 1)x^n-1) - nxⁿ
⇒ (1 - x)S = 1 + x + x² + x³ + ... + x^n-1 - nxⁿ

Now, RHS is a GP (except the last term) whose first term is 1 and common ratio is x.
∴ (1 - x)S = $\frac{{\mathrm{x}}^{\mathrm{n}}-1}{\mathrm{x}-1}$ - nxⁿ

⇒ (x - 1)S =  nxⁿ - $\frac{{\mathrm{x}}^{\mathrm{n}}-1}{\mathrm{x}-1}$

⇒ S =  $\frac{{\mathrm{nx}}^{\mathrm{n}}}{\mathrm{x}-1}$ - $\frac{{\mathrm{x}}^{\mathrm{n}}-1}{{\left(\mathrm{x}-1\right)}^2}$

Hence, option (c).

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