A sequence is an ordered list of numbers. A series is the sum of terms in a sequence. These ideas are key in algebra, finance, and modeling.

Sequence notation

Terms are often written \(u_1,u_2,u_3,\dots\) where \(u_n\) is the \(n\)-th term.

Example: \(2,5,8,11,\dots\) has \(u_1=2\), \(u_2=5\), \(u_3=8\).

Arithmetic sequences

Difference between consecutive terms is constant \(d\).

\[ u_n=a+(n-1)d \]

where \(a\) is first term.

Example: \(a=4,d=3\Rightarrow u_n=4+3(n-1)=3n+1\).

Arithmetic series

Sum of first \(n\) terms:

\[ S_n=\frac{n}{2}\big(2a+(n-1)d\big)=\frac{n}{2}(a+l) \]

where \(l\) is the \(n\)-th term.

Geometric sequences

Ratio between consecutive terms is constant \(r\).

\[ u_n=ar^{n-1} \]

Example: \(3,6,12,24,\dots\) has \(a=3,r=2\).

Geometric series

For \(r\ne1\), sum of first \(n\) terms:

\[ S_n=a\frac{1-r^n}{1-r} \]

Equivalent form: \(S_n=a\frac{r^n-1}{r-1}\).

Infinite geometric series

If \(|r|<1\), the sum to infinity exists:

\[ S_\infty=\frac{a}{1-r}. \]

If \(|r|\ge1\), no finite sum to infinity.

Sigma notation

\[ \sum_{k=1}^{n} f(k) \] means add values of \(f(k)\) from \(k=1\) to \(k=n\).

Example: \(\sum_{k=1}^{4}(2k+1)=3+5+7+9=24.\)

Typical problem-solving workflow

1. Identify arithmetic or geometric behavior.
2. Find key parameters (\(a,d\) or \(a,r\)).
3. Use \(u_n\) for term questions, \(S_n\) for sum questions.
4. Check reasonableness (growth/decay, sign, size).

Exam strategy

• Do not mix \(u_n\) and \(S_n\) formulas.
• For geometric sums, check \(r\ne1\) before applying formula.
• For infinity sums, always test \(|r|<1\).
• Define variables clearly before solving word problems.