Certainly! Let's solve this step-by-step.
We are given that the second term ([tex]\(a_2\)[/tex]) of the geometric progression (GP) is 48 and the fifth term ([tex]\(a_5\)[/tex]) is 6.
### Step 1: Find the first term and common ratio
Let's denote:
- First term as [tex]\(a\)[/tex]
- Common ratio as [tex]\(r\)[/tex]
#### Equations based on given information:
For the second term:
[tex]\[ a \cdot r = 48 \][/tex]
For the fifth term:
[tex]\[ a \cdot r^4 = 6 \][/tex]
To find [tex]\(r\)[/tex], we divide the fifth term equation by the second term equation:
[tex]\[
\frac{a \cdot r^4}{a \cdot r} = \frac{6}{48} \implies r^3 = \frac{1}{8}
\][/tex]
Solving for [tex]\(r\)[/tex]:
[tex]\[
r = \frac{1}{2}
\][/tex]
Now, using the value of [tex]\(r\)[/tex] to find [tex]\(a\)[/tex]:
[tex]\[
a \cdot \frac{1}{2} = 48 \implies a = 48 \cdot 2 = 96
\][/tex]
So, the first term ([tex]\(a\)[/tex]) is 96 and the common ratio ([tex]\(r\)[/tex]) is [tex]\(\frac{1}{2}\)[/tex].
### Step 2: Find the fourth term ([tex]\(a_4\)[/tex])
The nth term of a geometric progression is given by:
[tex]\[
a_n = a \cdot r^{n-1}
\][/tex]
For the fourth term:
[tex]\[
a_4 = a \cdot r^{4-1} = 96 \cdot \left(\frac{1}{2}\right)^3 = 96 \cdot \frac{1}{8} = 12
\][/tex]
So, the fourth term ([tex]\(a_4\)[/tex]) is 12.
### Step 3: Find the sum of the first 6 terms
The sum of the first [tex]\(n\)[/tex] terms ([tex]\(S_n\)[/tex]) of a geometric progression is given by:
[tex]\[
S_n = a \cdot \frac{1-r^n}{1-r} \quad \text{for } r < 1
\][/tex]
For the first 6 terms:
[tex]\[
S_6 = 96 \cdot \frac{1-\left(\frac{1}{2}\right)^6}{1-\frac{1}{2}} = 96 \cdot \frac{1-\frac{1}{64}}{\frac{1}{2}} = 96 \cdot \frac{\frac{63}{64}}{\frac{1}{2}} = 96 \cdot \frac{63}{32} = 96 \cdot 1.96875 = 189
\][/tex]
So, the sum of the first 6 terms ([tex]\(S_6\)[/tex]) is 189.
### Summary:
1. The first term ([tex]\(a\)[/tex]) is 96 and the common ratio ([tex]\(r\)[/tex]) is [tex]\(\frac{1}{2}\)[/tex].
2. The fourth term ([tex]\(a_4\)[/tex]) is 12.
3. The sum of the first 6 terms ([tex]\(S_6\)[/tex]) is 189.
