Summation Notation — Question 8

Question 8 - Solution

Let $$S=\sum_{k=0}^{n} r^k=1+r+r^2+\cdots+r^n.$$

Multiply both sides by $r$: $$rS=r+r^2+r^3+\cdots+r^{n+1}.$$

Subtract the second equation from the first: $$S-rS=(1+r+r^2+\cdots+r^n)-(r+r^2+\cdots+r^{n+1}).$$

All intermediate terms cancel, leaving $$S-rS=1-r^{n+1}.$$

Factor the left-hand side: $$(1-r)S=1-r^{n+1}.$$

Since $r\neq 1$, divide both sides by $1-r$: $$S=\frac{1-r^{n+1}}{1-r}.$$

Thus, $$\boxed{\sum_{k=0}^{n} r^k=\frac{1-r^{\,n+1}}{1-r}}$$