To prove that the nth term of an arithmetic progression of order k can be represented by the formula shown in the attached image, we can use the fact that the sum of a finite arithmetic progression is equal to the product of the number of terms and the average of the first and last terms.
First, let's define the arithmetic progression as {a, a+d, a+2d, ..., a+(n-1)d}, where a is the first term and d is the common difference between each term. We can also define the sum of the first n terms as S(n).
Now, the sum of the first n+1 terms can be represented as S(n+1) = a + (a+d) + (a+2d) + ... + (a+nd) = (n+1)a + d + 2d + ... + nd.
We can also use the fact that the sum of the first n terms is S(n) = a + (a+d) + (a+2d) + ... + (a+(n-1)d) = na + d + 2d + ... + (n-1)d.
Subtracting these two equations, we get S(n+1) - S(n) = (n+1)a + d + 2d + ... + nd - (na + d + 2d + ... + (n-1)d) = (n+1)a - na + d - d + 2d - 2d + ... + nd - (n-1)d = a + nd.
Now, we can rearrange this equation to get a + nd = S(n+1) - S(n). Substituting this into the formula for the nth term, we get the desired result: a + nd = (n+1)a - S(n+1) + S(n).
Therefore, we have proven that the nth term of an arithmetic progression of order k can be represented by the formula shown in the attached image. We used the fact that the sum of a finite arithmetic progression is equal to the product of the number of terms and the average of the first and last terms, and the fact that the expansion of (n+1) and S(n+1) - S(n) can be rearranged to get a + nd.
