In a geometric sequence, each term is given by the formula:

$T_n = ar^{n-1}$

where:

• $T_n$ is the $n$-th term,
• $a$ is the first term,
• $r$ is the common ratio, and
• $n$ is the term number.

Given:

• Third term $T_3 = 8$,
• Fifth term $T_5 = 128$.

a) Find the common ratio $r$ and the first term $a$.

We can use the formula for the terms of the geometric sequence.

For the third term: $T_3 = ar^2 = 8$

For the fifth term: $T_5 = ar^4 = 128$

Now, divide the equation for $T_5$ by the equation for $T_3$: $\frac{ar^4}{ar^2} = \frac{128}{8}$ $r^2 = 16$ $r = \pm 4$

Now, substitute $r = 4$ (we will first try the positive value) into the equation for $T_3$: $ar^2 = 8$ $a(4^2) = 8$ $16a = 8$ $a = \frac{8}{16} = \frac{1}{2}$

So, the first term $a = \frac{1}{2}$ and the common ratio $r = 4$.

b) Find the eighth term.

The eighth term $T_8$ is given by: $T_8 = ar^7$ Substitute $a = \frac{1}{2}$ and $r = 4$: $T_8 = \frac{1}{2} \times 4^7$ $T_8 = \frac{1}{2} \times 16384$ $T_8 = 8192$

Final Answers:

• The first term $a = \frac{1}{2}$,
• The common ratio $r = 4$,
• The eighth term $T_8 = 8192$.

Would you like more details or further clarifications?

Here are some related questions to consider:

1. How does the sign of $r$ affect the sequence?
2. What happens if the common ratio is negative?
3. Can you derive the formula for the sum of the first $n$ terms of a geometric sequence?
4. How would you handle this if given the sum of terms instead of individual terms?
5. What is the general behavior of a geometric sequence when $r > 1$ compared to when $0 < r < 1$?

Tip: In geometric sequences, knowing two terms can often help you solve for both the common ratio and the first term by setting up a system of equations.