Question: Write down the first four terms and the nth term.
2, 8, 32, 128
the nth term is 2 \times 4^{n-1}
The common ratio of a geometric sequence is 7. The 4th term is 1029.
What is the 3rd term?
What is the 1st term?
1. **State the problem:** We have a geometric sequence with common ratio $7$ and the 4th term is $1029$. We need to find the 3rd term and the 1st term.
2. **Recall the formula for the nth term of a geometric sequence:**
$$a_n = a_1 \times r^{n-1}$$
where $a_n$ is the nth term, $a_1$ is the first term, and $r$ is the common ratio.
3. **Use the given information for the 4th term:**
$$a_4 = a_1 \times 7^{4-1} = a_1 \times 7^3 = a_1 \times 343 = 1029$$
4. **Solve for the first term $a_1$:**
$$a_1 = \frac{1029}{343}$$
Show cancellation:
$$a_1 = \frac{\cancel{1029}}{\cancel{343}} = 3$$
5. **Find the 3rd term $a_3$ using the formula:**
$$a_3 = a_1 \times 7^{3-1} = 3 \times 7^2 = 3 \times 49 = 147$$
**Final answers:**
- The 1st term is $3$.
- The 3rd term is $147$.