Sequences & series

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.

History

Sequences and sums appear in ancient mathematics, including arithmetic progressions used for counting and trade. Later, geometric series became central in finance and calculus.

Derivation and reasoning

Why the arithmetic sum formula works

Write \(S_n=a+(a+d)+\dots+l\) and reverse it. Adding line-by-line gives \(2S_n=n(a+l)\), so \(S_n=\frac{n}{2}(a+l)\).

Why the geometric sum formula works

Let \(S_n=a+ar+\dots+ar^{n-1}\). Multiply by \(r\), subtract, and simplify: \[ (1-r)S_n=a(1-r^n). \]

Checkpoints

Find the 20th term of \(5,8,11,\dots\).

Answer: \(u_{20}=5+19(3)=62\).

Find \(S_{30}\) for \(2,5,8,\dots\).

Answer: \(a=2,d=3\Rightarrow S_{30}=\frac{30}{2}[2(2)+29(3)]=1365\).

For \(3,6,12,\dots\), find the 8th term.

Answer: \(u_8=3\cdot2^7=384\).

Compute \(1+\frac12+\frac14+\frac18+\dots\).

Answer: \(a=1,r=\frac12\Rightarrow S_\infty=\frac{1}{1-1/2}=2\).

Evaluate \(\sum_{k=1}^{5}(k+2)\).

Answer: \(3+4+5+6+7=25\).

Applications

• Savings plans: recurring deposits form sequences and cumulative sums.
• Population models: repeated growth can be modeled geometrically.
• Computer science: loops and algorithm costs often use summation notation.

References

• Standard high-school algebra curricula on arithmetic and geometric progressions.
• Precalculus texts on sigma notation and finite/infinite geometric series.