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

Well we need to first find the common ratio

\[\large -, 20, -, -, 2500 \]

Lets modify that a bit until we find the ratio, lets let 20 = a_1 and 2500 = a_4

\[\large a_n = a_1 \times r^{n-1}\]
\[\large 2500 = 20 \times r^{4 - 1}\]
\[\large 125 = r^3\]
\[\large r = 5\]

So we found the common ratio to be 5, we can now solve for the first term since we know each term is just multiplied by 5 to make the next term...

a_2 = 20...so a_1 = 20/5 = 4

...so our explicit rule will be

\[\large a_n = 4 \times 5^{n - 1}\]

Answer 2

thank you so much (:

Answer 3

Anytime :)