To find the terms of the arithmetic progression (AP) given the sum of the terms and the relationship between the terms, we will follow these steps:
1. Identify the form of the terms in the AP:
- Let the first term be [tex]\( a \)[/tex] and the common difference be [tex]\( d \)[/tex].
- The five terms of the AP can be expressed as: [tex]\( a \)[/tex], [tex]\( a + d \)[/tex], [tex]\( a + 2d \)[/tex], [tex]\( a + 3d \)[/tex], [tex]\( a + 4d \)[/tex].
2. Set up the sum equation for the AP:
- The sum of these five terms is given as 55.
- Using the sum formula for an arithmetic progression:
[tex]\[
\text{Sum} = \frac{n}{2} \times (\text{First term} + \text{Last term})
\][/tex]
- With [tex]\( n = 5 \)[/tex], the first term [tex]\( a \)[/tex], and the last term [tex]\( a + 4d \)[/tex]:
[tex]\[
\text{Sum} = \frac{5}{2} \times (a + (a + 4d)) = 55
\][/tex]
- Simplifying this equation:
[tex]\[
\frac{5}{2} \times (2a + 4d) = 55
\][/tex]
[tex]\[
5(a + 2d) = 55
\][/tex]
[tex]\[
a + 2d = 11 \quad \text{(1)}
\][/tex]
3. Set up the condition involving the fourth term:
- We are given that the fourth term ([tex]\( a + 3d \)[/tex]) is five more than the sum of the first two terms ([tex]\( a + (a + d) \)[/tex]):
[tex]\[
a + 3d = (a + (a + d)) + 5
\][/tex]
- Simplifying this equation:
[tex]\[
a + 3d = 2a + d + 5
\][/tex]
[tex]\[
a + 3d - 2a - d = 5
\][/tex]
[tex]\[
-a + 2d = 5
\][/tex]
[tex]\[
a - 2d = -5 \quad \text{(2)}
\][/tex]
4. Solve the simultaneous equations:
- From equations (1) and (2):
[tex]\[
a + 2d = 11
\][/tex]
[tex]\[
a - 2d = -5
\][/tex]
- Adding these two equations:
[tex]\[
(a + 2d) + (a - 2d) = 11 + (-5)
\][/tex]
[tex]\[
2a = 6
\][/tex]
[tex]\[
a = 3
\][/tex]
- Substitute [tex]\( a = 3 \)[/tex] back into equation (1):
[tex]\[
3 + 2d = 11
\][/tex]
[tex]\[
2d = 8
\][/tex]
[tex]\[
d = 4
\][/tex]
5. Determine the terms of the AP:
- Using [tex]\( a = 3 \)[/tex] and [tex]\( d = 4 \)[/tex]:
- First term: [tex]\( a = 3 \)[/tex]
- Second term: [tex]\( a + d = 3 + 4 = 7 \)[/tex]
- Third term: [tex]\( a + 2d = 3 + 8 = 11 \)[/tex]
- Fourth term: [tex]\( a + 3d = 3 + 12 = 15 \)[/tex]
- Fifth term: [tex]\( a + 4d = 3 + 16 = 19 \)[/tex]
Thus, the terms of the arithmetic progression are:
[tex]\[ \boxed{3, 7, 11, 15, 19} \][/tex]
