# A geometric sequence is defined recursively by #a_n = 5a_(n-1)#, the first term of the sequence is 0.45. What is the explicit formula for the nth?

##### 2 Answers

#### Explanation:

In a geometric series

As

Hence,

#### Explanation:

The recursive definition

#a_2 = 5 a_1#

#a_color(red)(3) = 5 a_2#

#= 5 * (5 a_1)#

#=5^2 a_1#

#= 5^((color(red)(3)-1)) a_1#

#a_color(red)(4) = 5 a_3#

#= 5 * (5^2 a_1)#

#= 5^3 a_1#

#= 5^((color(red)(4)-1)) a_1#

#...#

We can guess that

#a_color(red)(n) = 5^((color(red)(n)-1)) a_1#

and show that this is true by induction.

For

#a_1 = 5^((1-1)) a_1 = 5^0 a_1 = a_1# .

For all positive integers

#a_n = 5^((n-1)) a_1#

#= 5 * 5^((n-2)) a_1#

#= 5 * (5^(((n-1)-1)) a_1)#

#= 5 a_{n-1}# .

Since it is given that the first term (

#a_n = 5^((n-1)) (0.45)# .