Sure, let's find the 6th term of the given arithmetic progression (A.P.), whose terms are:
[tex]\[ \frac{2m+1}{m}, \frac{2m-1}{m}, \frac{2m-3}{m}, \ldots \][/tex]
### Step 1: Identify the first term
The first term [tex]\( a \)[/tex] of the sequence is:
[tex]\[ a = \frac{2m+1}{m} \][/tex]
### Step 2: Find the common difference [tex]\( d \)[/tex]
The common difference [tex]\( d \)[/tex] of an arithmetic progression can be found by subtracting the first term from the second term. So, let's find the second term first:
Second term:
[tex]\[ a_2 = \frac{2m-1}{m} \][/tex]
Now, the common difference [tex]\( d \)[/tex] is:
[tex]\[ d = a_2 - a = \frac{2m-1}{m} - \frac{2m+1}{m} \][/tex]
Simplifying [tex]\( d \)[/tex]:
[tex]\[ d = \frac{(2m-1) - (2m+1)}{m} = \frac{2m-1-2m-1}{m} = \frac{-2}{m} \][/tex]
[tex]\[ d = -2.0 \][/tex]
### Step 3: Determine the 6th term
The [tex]\( n \)[/tex]-th term of an arithmetic progression can be found using the formula:
[tex]\[ a_n = a + (n-1)d \][/tex]
For the 6th term ([tex]\( n = 6 \)[/tex]):
[tex]\[ a_6 = a + (6-1)d \][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( d \)[/tex]:
[tex]\[ a_6 = \frac{2m+1}{m} + 5 \cdot (-2) \][/tex]
### Step 4: Simplify the expression for the 6th term
Given that [tex]\( a \)[/tex] and [tex]\( d \)[/tex]:
[tex]\[ a = \frac{2m+1}{m} = 3.0 \][/tex]
[tex]\[ d = -2.0 \][/tex]
So we now have:
[tex]\[ a_6 = 3.0 + 5 \cdot (-2.0) \][/tex]
[tex]\[ a_6 = 3.0 - 10.0 \][/tex]
[tex]\[ a_6 = -7.0 \][/tex]
### Conclusion
The 6th term of the arithmetic progression is:
[tex]\[ \boxed{-7.0} \][/tex]
