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.