To solve for the [tex]\(15^{\text{th}}\)[/tex] term and the sum of the first 15 terms of an arithmetic progression (A.P.), we start with the information provided:
1. The sum of the first 3 terms of an A.P. is 48.
2. The sum of the first 5 terms of an A.P. is 110.
We denote the first term of the A.P. as [tex]\(a\)[/tex] and the common difference as [tex]\(d\)[/tex].
### Step 1: Express the given sums in terms of [tex]\(a\)[/tex] and [tex]\(d\)[/tex].
The sum of the first [tex]\(n\)[/tex] terms [tex]\(S_n\)[/tex] of an A.P. is given by:
[tex]\[
S_n = \frac{n}{2} [2a + (n-1)d]
\][/tex]
#### For the first 3 terms ([tex]\(n=3\)[/tex]):
[tex]\[
S_3 = \frac{3}{2} [2a + 2d] = 48
\][/tex]
[tex]\[
\frac{3}{2} (2a + 2d) = 48
\][/tex]
[tex]\[
3(a + d) = 48
\][/tex]
[tex]\[
a + d = 16 \quad \text{(Equation 1)}
\][/tex]
#### For the first 5 terms ([tex]\(n=5\)[/tex]):
[tex]\[
S_5 = \frac{5}{2} [2a + 4d] = 110
\][/tex]
[tex]\[
\frac{5}{2} (2a + 4d) = 110
\][/tex]
[tex]\[
5(a + 2d) = 110
\][/tex]
[tex]\[
a + 2d = 22 \quad \text{(Equation 2)}
\][/tex]
### Step 2: Solve the system of linear equations.
We now have the following system of equations:
[tex]\[
a + d = 16 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
a + 2d = 22 \quad \text{(Equation 2)}
\][/tex]
Subtract Equation 1 from Equation 2:
[tex]\[
(a + 2d) - (a + d) = 22 - 16
\][/tex]
[tex]\[
d = 6
\][/tex]
Substitute [tex]\(d = 6\)[/tex] back into Equation 1:
[tex]\[
a + 6 = 16
\][/tex]
[tex]\[
a = 10
\][/tex]
### Step 3: Find the [tex]\(15^{\text{th}}\)[/tex] term.
The [tex]\(n^{\text{th}}\)[/tex] term of an A.P. is given by:
[tex]\[
a_n = a + (n-1)d
\][/tex]
For the [tex]\(15^{\text{th}}\)[/tex] term ([tex]\(n=15\)[/tex]):
[tex]\[
a_{15} = 10 + (15-1) \cdot 6
\][/tex]
[tex]\[
a_{15} = 10 + 14 \cdot 6
\][/tex]
[tex]\[
a_{15} = 10 + 84
\][/tex]
[tex]\[
a_{15} = 94
\][/tex]
### Step 4: Find the sum of the first 15 terms.
The sum of the first [tex]\(n\)[/tex] terms ([tex]\(S_n\)[/tex]) is given by:
[tex]\[
S_n = \frac{n}{2} [2a + (n-1)d]
\][/tex]
For [tex]\(n = 15\)[/tex]:
[tex]\[
S_{15} = \frac{15}{2} [2 \cdot 10 + 14 \cdot 6]
\][/tex]
[tex]\[
S_{15} = \frac{15}{2} [20 + 84]
\][/tex]
[tex]\[
S_{15} = \frac{15}{2} \cdot 104
\][/tex]
[tex]\[
S_{15} = 15 \cdot 52
\][/tex]
[tex]\[
S_{15} = 780
\][/tex]
### Final Answer
The [tex]\(15^{\text{th}}\)[/tex] term of the A.P. is [tex]\(94\)[/tex].
The sum of the first 15 terms of the A.P. is [tex]\(780\)[/tex].
