To find the required values in this geometric progression, we'll start by using the properties of the sequence given the third and eighth terms.
### Step-by-Step Solution
#### (a) Common Ratio
We know that in a geometric progression (G.P.), each term is obtained by multiplying the previous term by a constant ratio [tex]\( r \)[/tex].
Given:
- The third term ([tex]\(a_3\)[/tex]) is 4
- The eighth term ([tex]\(a_8\)[/tex]) is [tex]\(\frac{1}{256}\)[/tex]
Let's denote the first term by [tex]\( a \)[/tex] and the common ratio by [tex]\( r \)[/tex]. Therefore, the general term of a G.P. can be written as:
[tex]\[ a_n = a \cdot r^{n-1} \][/tex]
For the third term ([tex]\(a_3\)[/tex]):
[tex]\[ a \cdot r^2 = 4 \quad \text{(1)} \][/tex]
For the eighth term ([tex]\(a_8\)[/tex]):
[tex]\[ a \cdot r^7 = \frac{1}{256} \quad \text{(2)} \][/tex]
To find the common ratio [tex]\( r \)[/tex], we divide equation (2) by equation (1):
[tex]\[
\frac{a \cdot r^7}{a \cdot r^2} = \frac{\frac{1}{256}}{4}
\][/tex]
Simplifying the fractions:
[tex]\[
r^5 = \frac{1}{256} \times \frac{1}{4}
\][/tex]
[tex]\[
r^5 = \frac{1}{1024}
\][/tex]
Taking the fifth root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \left(\frac{1}{1024}\right)^{1/5}
\][/tex]
This simplifies to:
[tex]\[
r \approx 0.25 \quad \text{(common ratio)}
\][/tex]
#### (b) First Term
Now, let's find the first term [tex]\( a \)[/tex].
Using equation (1):
[tex]\[
a \cdot r^2 = 4
\][/tex]
We already have [tex]\( r \approx 0.25 \)[/tex]. Squaring [tex]\( r \)[/tex]:
[tex]\[
r^2 = (0.25)^2 = 0.0625
\][/tex]
Substituting [tex]\( r^2 \)[/tex] into equation (1):
[tex]\[
a \cdot 0.0625 = 4
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{4}{0.0625} = 64
\][/tex]
So, the first term is:
[tex]\[
a = 64
\][/tex]
#### (c) Sum of the First and Second Terms
The first term [tex]\( a \)[/tex] is given, and the common ratio [tex]\( r \)[/tex] is known.
The sum of the first and second terms can be found as:
[tex]\[
a + a \cdot r
\][/tex]
Substituting the known values:
[tex]\[
64 + 64 \cdot 0.25
\][/tex]
[tex]\[
64 + 16 = 80
\][/tex]
So, the sum of the first and second terms is:
[tex]\[
80
\][/tex]
### Summary
Thus, the results are:
(a) The common ratio is [tex]\( 0.25 \)[/tex].
(b) The first term is [tex]\( 64 \)[/tex].
(c) The sum of the first and second terms is [tex]\( 80 \)[/tex].
