Question

Given that the geometric sequence has a₁ = 4 and r = 0.3, find the fifth term of the sequence. Round to the nearest hundredth.

Answer 1

To find the fifth term of a geometric sequence, we can use the formula for the \( n \)-th term of a geometric sequence:

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

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

In this case, we have:
- \( a_1 = 4 \)
- \( r = 0.3 \)
- \( n = 5 \)

Now, we can substitute these values into the formula to find \( a_5 \):

\[
a_5 = a_1 \cdot r^{(5-1)} = 4 \cdot (0.3)^{4}
\]

Calculating \( (0.3)^4 \):

\[
(0.3)^4 = 0.3 \cdot 0.3 \cdot 0.3 \cdot 0.3 = 0.0081
\]

Now substituting back to find \( a_5 \):

\[
a_5 = 4 \cdot 0.0081 = 0.0324
\]

Finally, rounding \( 0.0324 \) to the nearest hundredth gives:

\[
\boxed{0.03}
\]