Question

Find the 8th term of the geometric sequence 7, comma, 28, comma, 112, comma, dot dot dot7,28,112,...

Answer 1

To find the 8th term of the geometric sequence given by the terms 7, 28, 112, ..., we first need to identify the first term and the common ratio.

1. **Identify the first term (a)**:
The first term \( a \) is \( 7 \).

2. **Calculate the common ratio (r)**:
The common ratio \( r \) is calculated by dividing the second term by the first term:
\[
r = \frac{28}{7} = 4
\]
We can verify it with the next pair of terms:
\[
r = \frac{112}{28} = 4
\]

3. **General formula for the n-th term**:
The n-th term of a geometric sequence can be found using the formula:
\[
a_n = a \cdot r^{n-1}
\]

4. **Find the 8th term**:
Substitute \( n = 8 \) into the formula:
\[
a_8 = 7 \cdot 4^{8-1} = 7 \cdot 4^7
\]

5. **Calculate \( 4^7 \)**:
We can calculate \( 4^7 \):
\[
4^7 = (4^3)^2 \cdot 4 = 64^2 \cdot 4 = 4096 \cdot 4 = 16384
\]

6. **Calculate \( a_8 \)**:
Now substitute \( 4^7 \) back into the equation:
\[
a_8 = 7 \cdot 16384 = 114688
\]

Thus, the 8th term of the geometric sequence is \( \boxed{114688} \).