Question

The first term of a geometric sequence is 3. The second term of the geometric

sequence is StartFraction 3 over 2 Superscript 2 Baseline EndFraction.



Which expression represents the hundredth term of the geometric sequence?

Answer options with 4 options
A.
StartFraction 3 over 2 Superscript 99 Baseline EndFraction

B.
StartFraction 3 over 2 Superscript 100 Baseline EndFraction

C.
StartFraction 3 over 2 Superscript 198 Baseline EndFraction

D.
StartFraction 3 over 2 Superscript 200 Baseline EndFraction

Answer 1

In a geometric sequence, the \( n \)-th term can be represented by the formula:

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

where \( a_1 \) is the first term and \( r \) is the common ratio.

Given that the first term \( a_1 = 3 \) and the second term \( a_2 = \frac{3}{2^2} = \frac{3}{4} \), we can find the common ratio \( r \):

\[
a_2 = a_1 \cdot r \implies \frac{3}{4} = 3 \cdot r
\]
\[
r = \frac{\frac{3}{4}}{3} = \frac{1}{4}
\]

Now that we have the common ratio \( r = \frac{1}{4} \), we can find the expression for the hundredth term \( a_{100} \):

\[
a_{100} = a_1 \cdot r^{100-1} = 3 \cdot \left( \frac{1}{4} \right)^{99}
\]

This can also be expressed as:

\[
a_{100} = 3 \cdot \frac{1^{99}}{4^{99}} = 3 \cdot \frac{1}{4^{99}} = \frac{3}{4^{99}} = \frac{3}{(2^2)^{99}} = \frac{3}{2^{198}}
\]

Now rewriting the term we found:

\[
a_{100} = \frac{3}{2^{198}} = \frac{3}{2^{198}} = \frac{3}{2^{198}} \cdot 1 = \frac{3}{2^{198}}
\]

To match the answer options:

Since \( a_{100} = \frac{3}{2^{198}} \), and in the provided options, the equivalent term is:

- **Option C**: \(\frac{3}{2^{198}}\)

Therefore, the correct choice is:

**C. \(\frac{3}{2^{198}}\)**