A term u_n is one entry; a sum S_n is the total of the first n entries. They answer different questions. Write S_n=a+(a+d)+. s+[a+(n-1)d] and the same sum in reverse. Each paired column totals 2a+(n-1)d=a+u_n. There are n columns. 2S_n=n[2a+(n-1)d], giving S_n= n2[2a+(n-1)d]= n2(a+u_n). For 7,11,15,, u_10=7+9(4)=43, while S_10= 102(7+43)=250. If a total is given, the equation for n may be quadratic. Keep only positive integer values. For “first exceeds”, check the two adjacent integer totals. The progression formulas are supplied; choosing a term or a sum, and defining the first term, remain your decisions.

A term \(u_n\) is one entry; a sum \(S_n\) is the total of the first \(n\) entries. They answer different questions.

Write \(S_n=a+(a+d)+\cdots+[a+(n-1)d]\) and the same sum in reverse. Each paired column totals \(2a+(n-1)d=a+u_n\). There are \(n\) columns.

\(2S_n=n[2a+(n-1)d]\), giving \(S_n=\dfrac n2[2a+(n-1)d]=\dfrac n2(a+u_n)\).

For \(7,11,15,\ldots\), \(u_{10}=7+9(4)=43\), while \(S_{10}=\dfrac{10}{2}(7+43)=250\).

If a total is given, the equation for \(n\) may be quadratic. Keep only positive integer values. For “first exceeds”, check the two adjacent integer totals.

The progression formulas are supplied; choosing a term or a sum, and defining the first term, remain your decisions.

Adding an arithmetic progression

Adding an arithmetic progression