S_n adds terms 1 through n. The block from term p through term q is S_q-S_p-1. Subtraction cancels exactly the unwanted first p-1 terms. The block contains q-p+1 entries. Find the two endpoint terms and use count times average, or subtract the two partial sums. If a formula for S_n is given, use u_n=S_n-S_n-1. For u_n=2n+1, terms 4 through 8 are 9,11,13,15,17, with total 5(9+17)/2=65. Exam check: S_8-S_4 omits term 4. The required subtraction is S_8-S_3.

\(S_n\) adds terms 1 through \(n\). The block from term \(p\) through term \(q\) is \(S_q-S_{p-1}\).

Subtraction cancels exactly the unwanted first \(p-1\) terms. The block contains \(q-p+1\) entries.

Find the two endpoint terms and use count times average, or subtract the two partial sums. If a formula for \(S_n\) is given, use \(u_n=S_n-S_{n-1}\).

For \(u_n=2n+1\), terms 4 through 8 are \(9,11,13,15,17\), with total \(\dfrac{5(9+17)}{2}=65\).

Exam check: \(S_8-S_4\) omits term 4. The required subtraction is \(S_8-S_3\).

Partial totals and blocks

Partial totals and blocks