Question

how many terms are in the following sequence:1, 5, 5^2, 5^3, ... 5^30100, 200, 300, 400, ...., 500001, 3, 9, 27, 81, 243, .... 6561

Answer 1

SOLUTION:

Step 1 :

The sequence is a good example of a Geometric Sequence

where

$\begin{gathered} l=ar^{n-1} \\ \text{where l = last term = 5}^{30} \\ a\text{ = first term = 5} \\ n\text{ =number of terms = ?} \\ \text{r = common ratio = }\frac{5^2}{5}\text{ = 5} \end{gathered}$

Step 2 :

Putting the values in the equation:

$\begin{gathered} 5^{30}\text{ = 5 ( }5)^{n-1} \\ \text{5 }^{30}=5^{1\text{ + n-1 }}=5^n \\ 5^{30\text{ }}=5^n \\ \text{equate the indices, we have that:} \\ n\text{ = 30} \end{gathered}$

CONCLUSION :

The number of terms, n = 30.