Since the terms are in an Arithmetic Progression (A.P.), the difference between consecutive terms should be equal. Let's denote the terms as follows:
1. [tex]\( 2k + 3 \)[/tex]
2. [tex]\( 3k - 1 \)[/tex]
3. [tex]\( 5k - 3 \)[/tex]
For terms to be in A.P., the difference between the second term and the first term should equal the difference between the third term and the second term. Thus, we set up the equation:
[tex]\[
(3k - 1) - (2k + 3) = (5k - 3) - (3k - 1)
\][/tex]
Let's simplify both sides of the equation step-by-step.
1. Simplifying the left-hand side:
[tex]\[
(3k - 1) - (2k + 3) = 3k - 1 - 2k - 3 = k - 4
\][/tex]
2. Simplifying the right-hand side:
[tex]\[
(5k - 3) - (3k - 1) = 5k - 3 - 3k + 1 = 2k - 2
\][/tex]
Now we have the equation:
[tex]\[
k - 4 = 2k - 2
\][/tex]
To solve for [tex]\( k \)[/tex], we manipulate the equation to isolate [tex]\( k \)[/tex] on one side:
[tex]\[
k - 4 = 2k - 2
\][/tex]
Subtract [tex]\( k \)[/tex] from both sides:
[tex]\[
-4 = k - 2
\][/tex]
Add 2 to both sides:
[tex]\[
-2 = k
\][/tex]
So, the value of [tex]\( k \)[/tex] is [tex]\( -2 \)[/tex].
Now we substitute [tex]\( k = -2 \)[/tex] back into the original terms to find the actual values of the terms in the A.P.:
1. First term: [tex]\( 2k + 3 \)[/tex]
[tex]\[
2(-2) + 3 = -4 + 3 = -1
\][/tex]
2. Second term: [tex]\( 3k - 1 \)[/tex]
[tex]\[
3(-2) - 1 = -6 - 1 = -7
\][/tex]
3. Third term: [tex]\( 5k - 3 \)[/tex]
[tex]\[
5(-2) - 3 = -10 - 3 = -13
\][/tex]
Therefore, the three terms of the arithmetic progression are:
[tex]\[
\boxed{-1, -7, -13}
\][/tex]
