Part 1 Show that for all positive integers \( n \), \( x \left[ (1+x)x^{n-1} + (1+x)^{n-2} + \cdots + (1+x)^2 + (1+x) +1 \right] = (1+x)^n -1 \). \( \begin{align} \displaystyle \text{LHS} &= x \left[(1+x)x^{n-1} + (1+x)^{n-2} + \cdots + (1+x)^2 + (1+x) +1 \right] \\ &= x \times \frac{1\left[ (1+x)^n – 1 \right]}{(1+x)-1} […]
