Sequences and Series
1. Arithmetic sequences and series
Arithmetic structure is additive: equal jumps between consecutive terms.
nth term
\[u_n=a+(n-1)d\]finite sum
\[S_n=\frac n2[2a+(n-1)d]=\frac n2(a+l)\]Find parameters
If \(u_5=18\) and \(u_{12}=46\), then \[a+4d=18,\qquad a+11d=46.\] Subtracting gives \(7d=28\Rightarrow d=4\), then \(a=2\).Exam tip. Write the correct term equation before substituting numbers. Most AP errors come from using \(a+nd\) instead of \(a+(n-1)d\).
2. Geometric sequences and series
Geometric structure is multiplicative: each term is obtained by a constant ratio.
nth term
\[u_n=ar^{n-1}\]finite sum
\[S_n=\frac{a(1-r^n)}{1-r}\quad(r
e1)\]sum to infinity
\[S_\infty=\frac a{1-r}\quad\text{only if }|r|<1\]Infinite series
For \(6-3+1.5-\cdots\), \(a=6\) and \(r=-\frac12\). Since \(|r|<1\), \[S_\infty=\frac6{1+1/2}=4.\]Common mistake. Never use \(S_\infty=\frac a{1-r}\) without checking \(|r|<1\).
3. Sigma notation and mixed AP/GP problems
Sigma questions test whether you recognise the underlying sequence before calculating.
Meaning
\[\sum_{r=1}^{n}f(r)=f(1)+f(2)+\cdots+f(n)\]Linearity
\[\sum(af(r)+bg(r))=a\sum f(r)+b\sum g(r)\]Convert sigma to a known series
\[\sum_{r=1}^{20}(3r+2)=3\sum_{r=1}^{20}r+2(20).\] Using \(\sum_{r=1}^{n}r=\frac{n(n+1)}2\): \[3\cdot\frac{20\cdot21}{2}+40=670.\]Find number of terms
If an AP has \(a=7,d=5\) and final term \(l=102\), solve \[102=7+(n-1)5\Rightarrow n=20.\] Then choose the most convenient sum formula.4. Modelling and equation setup
Long word problems usually reduce to identifying whether change is additive (AP) or multiplicative (GP).
- Define the first term and common difference/ratio from the context.
- Translate the stated term or total into \(u_n\) or \(S_n\).
- Solve for unknown parameters, including \(n\) if needed.
- Check that \(n\) is a positive integer and that any infinite-GP condition is valid.
Exam tip. When \(n\) appears both outside and in an exponent through \(r^n\), logarithms may be needed. Keep the exact model before using a calculator.