### Math Notes

Subjects

#### Algebra Solutions

##### Topics || Problems

Derivation: Summation of Geometric Series

Let r be the common ratio of the series where | r | < 1, a1 be the first term and an be the the nth term.

S = a1 + a2 + a3 + ... + an

$$r = \frac{a_2}{a_1} = \frac{a_3}{a_2}$$

S = a1 + r a1 + r2a1 + r3a1+... + rn-1a1

$$\frac{S}{a_1} = 1 + r + r^2 +r^3 + ... + r^{n-1}$$

$$\frac{rS}{a_1} = r + r^2 + r^3 +r^4 + ... + r^{n-1}+ r^{n}$$

$$\frac{rS}{a_1} =( \frac{S}{a_1} - 1)+ r^{n}$$

$$\frac{rS}{a_1} - \frac{S}{a_1} = r^{n} - 1$$

$$\frac{rS-S}{a_1} = r^{n}$$-1

$$\frac{S(r-1)}{a_1} = r^{n}$$-1

$$S = a_1 \frac{r^{n}-1}{r-1}$$

OR

$$S = a_1 \frac{1-r^{n}}{1-r}$$