Core rules and standard sums

Use integer bounds \(m\le n\) for the displayed finite sums; \(n\) is a positive integer in the standard sums and geometric sum. In the final formula using \(F(m-1)\), take \(1\le m\le n\) and \(F(0)=0\).

Linearity

\[\sum_{r=m}^{n}(au_r+bv_r)=a\sum_{r=m}^{n}u_r+b\sum_{r=m}^{n}v_r\]

\[\sum_{r=m}^{n}c=(n-m+1)c\]

Standard sums from \(1\) to \(n\)

\[\sum_{r=1}^{n}r=\frac{n(n+1)}2\]

\[\sum_{r=1}^{n}r^2=\frac{n(n+1)(2n+1)}6\]

\[\sum_{r=1}^{n}r^3=\left[\frac{n(n+1)}2\right]^2\]

Geometric sums

\[\sum_{r=0}^{n-1}aq^r=\frac{a(1-q^n)}{1-q}\quad(q\ne1)\]

A sum that starts later

If \(F(n)=\sum_{r=1}^{n}u_r\), then

\[\sum_{r=m}^{n}u_r=F(n)-F(m-1).\]

IB AA scope: sigma notation and arithmetic/geometric sums are prescribed. The square-sum and cube-sum identities here are useful results to derive or supply when needed; they are not additional AA formula-memorisation requirements.