Question

What is the 8th term of a geometric sequence for which a1=4
and a2=20?

a8=

Answer 1

In a geometric sequence, each term can be expressed in terms of the previous term by a common ratio \( r \). The first term \( a_1 \) and the second term \( a_2 \) are related by the formula:

\[
a_2 = a_1 \cdot r
\]

We know that \( a_1 = 4 \) and \( a_2 = 20 \). Plugging in these values, we get:

\[
20 = 4 \cdot r
\]

To find \( r \), we can rearrange the equation:

\[
r = \frac{20}{4} = 5
\]

Now that we have the common ratio \( r = 5 \), we can find the \( n \)-th term of the geometric sequence using the formula:

\[
a_n = a_1 \cdot r^{n-1}
\]

To find the 8th term \( a_8 \):

\[
a_8 = a_1 \cdot r^{8-1} = 4 \cdot 5^7
\]

Now we calculate \( 5^7 \):

\[
5^1 = 5 \\
5^2 = 25 \\
5^3 = 125 \\
5^4 = 625 \\
5^5 = 3125 \\
5^6 = 15625 \\
5^7 = 78125
\]

Substituting this back into the formula for \( a_8 \):

\[
a_8 = 4 \cdot 78125 = 312500
\]

Therefore, the 8th term of the geometric sequence is:

\[
\boxed{312500}
\]