Let $x = 1.1$. $S = 1 + 2x + 3x^2 + \ldots + 10x^9$ and $xS = x + 2x^2 + \ldots + 9x^9 + 10x^{10}$. Subtracting we get $S(1-x) = (1 + x + x^2 + \ldots + x^9) - 10x^{10}$. Find $S$.
Step-by-Step Solution
Key Concept: Use the standard technique of multiplying a series by $x$ and subtracting to telescope the coefficients.
Let $x = 1.1$. We have $S = 1 + 2x + 3x^2 + \ldots + 10x^9$. Multiplying by $x$: $xS = x + 2x^2 + 3x^3 + \ldots + 10x^{10}$. Subtracting: $S(1-x) = 1 + x + x^2 + \ldots + x^9 - 10x^{10}$. The sum $1 + x + x^2 + \ldots + x^9 = \frac{1-x^{10}}{1-x}$, so $S = \frac{\frac{1-x^{10}}{1-x} - 10x^{10}}{1-x} = \frac{1-x^{10} - 10x^{10}(1-x)}{(1-x)^2} = \frac{(1.1)^{10} - 100 \times (1.1)^{10} - 100}{(1-1.1)^2} = 100$.
Correct Answer: 100