Question

Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are 20 and 2500, respectively.

Answer 1

The explicit formula of the geometric sequence is:

`a_n=4* 5^(n-1)`

Explanation:

The explicit formula is the expression where the nth term is given in terms of the first term of the sequence.

We know that the explicit formula for a geometric sequence is given by:

`a_n=(a_1)^(r-1)`

Here we are given:

The second and fifth terms as: 20 and 2500 respectively.

i.e.

`a_2=20` and
`a_5=2500`

i.e.

`ar=20\ and\ ar^4=2500`

Hence,

`(ar)/(ar^4)=(20)/(2500)\\\\\\(1)/(r^3)=(1)/(125)\\\\\\((1)/(r))^3=((1)/(5))^3\\\\\\(1)/(r)=(1)/(5)\\\\\\r=5`

Also,

we have:

`ar=20\\\\i.e.\\\\a* 5=20\\\\i.e.\ a=4`

Hence, the explicit formula is given by:

`a_n=4* 5^(n-1)`

Answer 2

Given that the 2nd and 5th term of a geometric sequence is 20 and 2500, the formula will be obtained as follows.
The formula for geometric sequence is given by:
nth=ar^(n-1)
where:
a=1st term
r=common ratio
n=nth term
thus the 2nd term is:
20=ar^(2-1)
20=ar......i

the 5th term is:
2500=ar^(5-1)
2500=ar^4.......ii
from i
a=20/r

from ii
a=2500/r^4
therefore:
20/r=2500/r^4
multiplying through by r^4 we get:
20r^3=2500
dividing both sides by 20 we get:
r^3=125
hence;
r=5
substituting the value of r in i we get:
20=ar
20=5a
thus;
a=4
the formula for the sequence will therefore be:
nth=ar^(n-1)
nth=4*5^(n-1)