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------------------------------------------------ The first term of an Arithmetic Progression (AP) is 3, and its last term is 47. If the sum of the AP is 575, calculate the common difference.

To solve this problem, we need to find the common difference of an arithmetic progression (AP) given that the first term is 3, the last term is 47, and the sum of the series is 575.

Let's break it down step-by-step:

1. Understand the Sum of an AP:
The formula for the sum of an arithmetic progression is given by:
[tex]\[
S_n = \frac{n}{2} \times (\text{first term} + \text{last term})
\][/tex]
where [tex]$ S_n $[/tex] is the sum of the first [tex]$ n $[/tex] terms of the AP.

2. Substitute the Known Values:
We know:
- First term ([tex]$ a $[/tex]) = 3
- Last term ([tex]$ l $[/tex]) = 47
- Sum ([tex]$ S_n $[/tex]) = 575

Plug these values into the formula:
[tex]\[
575 = \frac{n}{2} \times (3 + 47)
\][/tex]
[tex]\[
575 = \frac{n}{2} \times 50
\][/tex]

3. Solve for [tex]$ n $[/tex]:
Multiply both sides by 2 to clear the fraction:
[tex]\[
1150 = 50n
\][/tex]
Divide by 50:
[tex]\[
n = \frac{1150}{50} = 23
\][/tex]

So, the number of terms ([tex]$ n $[/tex]) is 23.

4. Find the Common Difference ([tex]$ d $[/tex]):
The formula for the [tex]$ nth $[/tex] term of an AP is:
[tex]\[
l = a + (n-1)d
\][/tex]
We know:
- Last term ([tex]$ l $[/tex]) = 47
- First term ([tex]$ a $[/tex]) = 3
- Number of terms ([tex]$ n $[/tex]) = 23

Substitute these values:
[tex]\[
47 = 3 + (23-1)d
\][/tex]
Simplify:
[tex]\[
47 = 3 + 22d
\][/tex]
Subtract 3 from both sides:
[tex]\[
44 = 22d
\][/tex]
Divide by 22:
[tex]\[
d = \frac{44}{22} = 2
\][/tex]

Therefore, the common difference of the arithmetic progression is 2.