Question

Find the 8th term of the geometric sequence 4, comma, minus, 20, comma, 100, comma, dot dot dot4,−20,100,...

Answer 1

To solve for the 8th term of the geometric sequence given by \( 4, -20, 100, \ldots \), we first need to find the common ratio.

In a geometric sequence, the common ratio \( r \) is found by dividing any term by the previous term. Let's calculate the common ratio using the first two terms:

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

Next, we can verify this ratio by checking the second and third terms:

\[
r = \frac{100}{-20} = -5
\]

Now that we have confirmed the common ratio \( r = -5 \), we can use the formula for the \( n \)-th term of a geometric sequence, which is given by:

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

where:
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.

For this sequence:
- \( a_1 = 4 \)
- \( r = -5 \)

We need to find the 8th term (\( n = 8 \)):

\[
a_8 = 4 \cdot (-5)^{8-1} = 4 \cdot (-5)^7
\]

Now, we calculate \( (-5)^7 \):

\[
(-5)^7 = -5^7 = -78125 \quad (\text{since } 5^7 = 78125)
\]

Now we can compute \( a_8 \):

\[
a_8 = 4 \cdot (-78125) = -312500
\]

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

\[
\boxed{-312500}
\]