Q. 90 - Calculus

Question

Prove that the ratio of successive terms of a nonzero geometric sequence is constant

Step-by-Step Solution

Answer

Proved

Step 1. Given

Consider the geometric sequence $\left\{a_k\right\}=\left\{c{r}^k\right\}$

Step 2. Proof

The general term of the sequence is $\left\{a_k\right\}=\left\{c{r}^k\right\}$ is $a_k=c{r}^k$.

The ratio of two successive terms is $\left\{a_k\right\}=\left\{c{r}^k\right\}$ is:

$$\begin{gathered} \frac{a_{k+1}}{a_k}=\frac{c{r}^{k+1}}{c{r}^k} \\ =r^{k+1-k} \\ =r \\ \mathrm{Thus} \mathrm{the} \mathrm{ratio} \frac{a_{k+1}}{a_k} \mathrm{is} \mathrm{constant}. \end{gathered}$$