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

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)