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
\[\sum_{r=m}^{n}u_r=F(n)-F(m-1).\]
Decision order: expand or simplify the general term → split the sum → apply standard formulae → simplify at the end.

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

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

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

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

Decision order: expand or simplify the general term → split the sum → apply standard formulae → simplify at the end.

Mathematics

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Core rules and standard sums

Core rules and standard sums