Question

Given the following geometric sequence, find the terms
3,21
3rd term:
5th term:
7th term:

Answer 1

In a geometric sequence, each term after the first is found by multiplying the previous term by a common ratio, denoted as \( r \).

Given the first two terms of the sequence:
- \( a_1 = 3 \)
- \( a_2 = 21 \)

To find the common ratio \( r \):
\[
r = \frac{a_2}{a_1} = \frac{21}{3} = 7
\]

Now we can express the general term \( a_n \) of the geometric sequence:
\[
a_n = a_1 \cdot r^{(n-1)} = 3 \cdot 7^{(n-1)}
\]

Now we can calculate the required terms:

1. **3rd term (\( a_3 \))**:
\[
a_3 = 3 \cdot 7^{(3-1)} = 3 \cdot 7^2 = 3 \cdot 49 = 147
\]

2. **5th term (\( a_5 \))**:
\[
a_5 = 3 \cdot 7^{(5-1)} = 3 \cdot 7^4 = 3 \cdot 2401 = 7203
\]

3. **7th term (\( a_7 \))**:
\[
a_7 = 3 \cdot 7^{(7-1)} = 3 \cdot 7^6 = 3 \cdot 117649 = 352947
\]

Thus, the calculated terms are:
- **3rd term**: 147
- **5th term**: 7203
- **7th term**: 352947