Question

Given the recursive formula for a geometric sequence find the common ratio the first five terms and the explicit formula

Answer 1

First, let find the first five terms using the recursive formula:

Then:

$a_n=a_{n-1}\cdot2$

Now,

a1 = 2

For a2 :

$a_2=a_{2-1}\cdot2=a_1\cdot2=2\cdot2=4$

a2= 4

For a3:

$a_3=a_{3-1}\cdot2=a_2\cdot2=4\cdot2=8$

a3 = 8

For a4:

$a_4=a_{4-1}\cdot2=a_3\cdot2=8\cdot2=16$

a4 = 16

For a5:

$a_5=a_{5-1}\cdot2=a_4\cdot2=16\cdot2=32$

a5 = 32

Therefore, the five first terms are 2,4,8,16,32.

Now, the common ratio is 2.

The explicit formula hast the next form:

$a_n=a_1+(n-1)d$

where d is the common ratio.

Replace using a1=2 and d=2

Therefore, the explicit formula is given by:

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