Question

Write an explicit formula for the geometric sequence 3, 6, 12, 24, 48, 96.. Use your explicit formula to find the 100th term of your sequence - your number may be large, make sure you use correct Scientific Notation.

Answer 1

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio.

Given the sequence: 3, 6, 12, 24, 48, 96, we can identify the first term and the common ratio.

1. The first term \( a_1 \) is 3.
2. To find the common ratio \( r \), we divide the second term by the first term:
\[
r = \frac{6}{3} = 2
\]

The explicit formula for the \( n \)-th term of a geometric sequence can be expressed as:
\[
a_n = a_1 \cdot r^{n-1}
\]
Substituting the values we have:
\[
a_n = 3 \cdot 2^{n-1}
\]

Now, to find the 100th term (\( a_{100} \)):
\[
a_{100} = 3 \cdot 2^{100 - 1} = 3 \cdot 2^{99}
\]

Next, we need to compute \( 2^{99} \).

Calculating \( 2^{99} \):
\[
2^{99} = 2^{10} \cdot 2^{89} = 1024 \cdot 2^{89}
\]

We can use logarithms or a calculator to find \( 2^{99} \):
\[
2^{99} \approx 6.338253 \times 10^{29}
\]

Now multiplying by 3:
\[
a_{100} = 3 \cdot 2^{99} \approx 3 \cdot 6.338253 \times 10^{29} \approx 1.901476 \times 10^{30}
\]

Thus, the explicit formula for the sequence is:
\[
a_n = 3 \cdot 2^{n-1}
\]

And the 100th term, in scientific notation, is approximately:
\[
a_{100} \approx 1.901476 \times 10^{30}
\]