Answer :
To solve the problem of finding the common difference in an Arithmetic Progression (AP), let's break it down step-by-step:
1. Understand the problem:
- First term ([tex]\(a\)[/tex]) of the AP is 3.
- Last term ([tex]\(l\)[/tex]) of the AP is 47.
- The sum ([tex]\(S_n\)[/tex]) of the AP is 575.
- We need to find the common difference ([tex]\(d\)[/tex]).
2. Use the formula for the sum of an AP:
[tex]\[
S_n = \frac{n}{2} \times (a + l)
\][/tex]
Where:
- [tex]\(S_n\)[/tex] is the sum of the first [tex]\(n\)[/tex] terms.
- [tex]\(n\)[/tex] is the number of terms.
- [tex]\(a\)[/tex] is the first term.
- [tex]\(l\)[/tex] is the last term.
3. Substitute known values into the formula:
[tex]\[
575 = \frac{n}{2} \times (3 + 47)
\][/tex]
Simplify the expression:
[tex]\[
575 = \frac{n}{2} \times 50
\][/tex]
4. Solve for [tex]\(n\)[/tex]:
Multiply both sides by 2 to clear the fraction:
[tex]\[
1150 = 50n
\][/tex]
Divide both sides by 50 to solve for [tex]\(n\)[/tex]:
[tex]\[
n = \frac{1150}{50} = 23
\][/tex]
The number of terms ([tex]\(n\)[/tex]) in the AP is 23.
5. Use the formula for the last term of an AP to find the common difference ([tex]\(d\)[/tex]):
[tex]\[
l = a + (n-1) \times d
\][/tex]
Substitute the known values:
[tex]\[
47 = 3 + (23-1) \times d
\][/tex]
Simplify to solve for [tex]\(d\)[/tex]:
[tex]\[
47 = 3 + 22d
\][/tex]
Subtract 3 from both sides:
[tex]\[
44 = 22d
\][/tex]
Divide both sides by 22:
[tex]\[
d = \frac{44}{22} = 2
\][/tex]
Thus, the common difference ([tex]\(d\)[/tex]) in the Arithmetic Progression is 2.
1. Understand the problem:
- First term ([tex]\(a\)[/tex]) of the AP is 3.
- Last term ([tex]\(l\)[/tex]) of the AP is 47.
- The sum ([tex]\(S_n\)[/tex]) of the AP is 575.
- We need to find the common difference ([tex]\(d\)[/tex]).
2. Use the formula for the sum of an AP:
[tex]\[
S_n = \frac{n}{2} \times (a + l)
\][/tex]
Where:
- [tex]\(S_n\)[/tex] is the sum of the first [tex]\(n\)[/tex] terms.
- [tex]\(n\)[/tex] is the number of terms.
- [tex]\(a\)[/tex] is the first term.
- [tex]\(l\)[/tex] is the last term.
3. Substitute known values into the formula:
[tex]\[
575 = \frac{n}{2} \times (3 + 47)
\][/tex]
Simplify the expression:
[tex]\[
575 = \frac{n}{2} \times 50
\][/tex]
4. Solve for [tex]\(n\)[/tex]:
Multiply both sides by 2 to clear the fraction:
[tex]\[
1150 = 50n
\][/tex]
Divide both sides by 50 to solve for [tex]\(n\)[/tex]:
[tex]\[
n = \frac{1150}{50} = 23
\][/tex]
The number of terms ([tex]\(n\)[/tex]) in the AP is 23.
5. Use the formula for the last term of an AP to find the common difference ([tex]\(d\)[/tex]):
[tex]\[
l = a + (n-1) \times d
\][/tex]
Substitute the known values:
[tex]\[
47 = 3 + (23-1) \times d
\][/tex]
Simplify to solve for [tex]\(d\)[/tex]:
[tex]\[
47 = 3 + 22d
\][/tex]
Subtract 3 from both sides:
[tex]\[
44 = 22d
\][/tex]
Divide both sides by 22:
[tex]\[
d = \frac{44}{22} = 2
\][/tex]
Thus, the common difference ([tex]\(d\)[/tex]) in the Arithmetic Progression is 2.
