Finite Arithmetic Progression
An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the “common difference,” denoted by (d). A finite arithmetic progression is simply an AP that has a fixed number of terms, (n). The first term is denoted as (a).
The (n)th term (T_n) of an AP can be calculated using the formula:
[
T_n = a + (n  1) \times d
]
The sum (S) of the first (n) terms in an arithmetic sequence can be calculated using the formula:
[
S = \frac{n}{2} \times (a + T_n)
]
or equivalently,
[
S = \frac{n}{2} \times (2a + (n  1) \times d)
]
Example Code
Java
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 public class ArithmeticProgression {
public static void main(String[] args) {
int a = 1; // First term
int d = 2; // Common difference
int n = 5; // Number of terms
// Calculate nth term and sum
int Tn = a + (n  1) * d;
int sum = n * (a + Tn) / 2;
System.out.println("The nth term is: " + Tn);
System.out.println("The sum of the first n terms is: " + sum);
}
}

C++
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 #include <iostream>
using namespace std;
int main() {
int a = 1; // First term
int d = 2; // Common difference
int n = 5; // Number of terms
// Calculate nth term and sum
int Tn = a + (n  1) * d;
int sum = n * (a + Tn) / 2;
cout << "The nth term is: " << Tn << endl;
cout << "The sum of the first n terms is: " << sum << endl;
return 0;
}

Python
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 # First term
a = 1
# Common difference
d = 2
# Number of terms
n = 5
# Calculate nth term and sum
Tn = a + (n  1) * d
sum = n * (a + Tn) // 2
print(f"The nth term is: {Tn}")
print(f"The sum of the first n terms is: {sum}")

In these example codes, the formula for the (n)th term and the sum of the first (n) terms in a finite arithmetic progression are implemented in Java, C++, and Python.