For an AP, each given term is an equation u_k=a+(k-1)d. Subtracting term equations removes a. There are q-p equal steps from term p to term q, so u_q-u_p=(q-p)d. Write both equations; subtract to find d; substitute to find a; check both given values. A supplied sum instead gives S_m= m2[2a+(m-1)d]. If u_3=11 and u_7=23, then 4d=12, d=3, and a=11-2(3)=5. Exam check: the difference between term 3 and term 7 spans four steps, not seven. Negative d is allowed.

For an AP, each given term is an equation \(u_k=a+(k-1)d\). Subtracting term equations removes \(a\).

There are \(q-p\) equal steps from term \(p\) to term \(q\), so \(u_q-u_p=(q-p)d\).

Write both equations; subtract to find \(d\); substitute to find \(a\); check both given values. A supplied sum instead gives \(S_m=\frac m2[2a+(m-1)d]\).

If \(u_3=11\) and \(u_7=23\), then \(4d=12\), \(d=3\), and \(a=11-2(3)=5\).

Exam check: the difference between term 3 and term 7 spans four steps, not seven. Negative \(d\) is allowed.

Recovering arithmetic parameters

Recovering arithmetic parameters