1
Generate, classify and sum sequences
Run an explicit or recursive definition, re-index a finite sum, distinguish arithmetic from geometric data, and derive the appropriate term and finite-sum formula.
Course · Algebra
Sequences and Series
A sequence is a function whose input is a position. Arithmetic and geometric families make that idea calculable: both have closed forms for a term and a finite sum, some geometric sums have a limit, and the same indexed notation leads to Pascal's triangle and binomial expansion.
aₙ = a₁ + (n − 1)d aₙ = a₁rⁿ⁻¹
Sₙ = n(a₁ + aₙ)/2 Sₙ = a₁(1 − rⁿ)/(1 − r)
S∞ = a₁/(1 − r) a₁ ≠ 0 needs |r| < 1; the zero series sums to 0
(a + b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ
What this course is about
- Subject
- Algebra
- Lessons
- 11
- Format
- Interactive
Overview
What you will be able to do
By the end you can generate and classify sequences, compute finite and infinite geometric sums under their conditions, apply them to decimals and payments, and construct the coefficients and terms of a binomial expansion.
2
Decide and apply geometric convergence
Use the sequence of partial sums and |r| < 1 under the nonzero-first-term condition, quantify the remaining gap, and convert a repeating decimal to an exact fraction.
3
Value a stream of payments
Place every ordinary-annuity payment on one timeline, then compute and check its accumulated value, present value or perpetuity value with rate and period in the same units.
4
Construct and use binomial coefficients
Build and check a row of Pascal's triangle, expand a signed binomial power, and isolate a requested term by solving its exponent equation for an admissible k.
Lessons
11 lessons
Sequences are generated and finite sums encoded first. Next the two standard families are classified and partial sums recovered or telescoped. Then geometric convergence is decided and applied to decimals and payment streams; last, binomial coefficients are constructed and used.
Sequences and RecursionGenerate a sequence from either definition and diagnose what an incomplete recursion is missing.Not marked
Sigma NotationExpand, encode and re-index a finite sum without changing any of its terms.Not marked
Arithmetic Sequences and SeriesClassify an arithmetic sequence, compute a requested term or sum, and justify the sum by pairing.Not marked
Geometric Sequences and SeriesClassify a geometric sequence and compute any requested term or finite sum, including the r = 1 case.Not marked
Partial Sums and TelescopingRecover terms from partial sums and collapse a telescoping sum only after verifying its split.Not marked
Infinite Geometric SeriesDecide whether a geometric series converges and quantify the exact gap after a stated number of terms.Not marked
Repeating Decimals as SeriesConvert a repeating decimal to a fraction and defend why its partial sums name that exact value.Not marked
Annuities and Accumulated PaymentsPlace an ordinary annuity on a timeline and compute its accumulated, present or perpetual value.Not marked
Pascal's TriangleConstruct and check a row of Pascal's triangle, then interpret one entry as a choice count.Not marked
The Binomial TheoremExpand a signed binomial power from its coefficients and verify the complete result by substitution.Not marked
The General Term of an ExpansionIsolate a requested term by solving for k, or prove that the expansion contains no such term.Not marked
Recommended background
Exponents, functions, and exact fractions.
Practice suggestions
After studying each complete worked example, cover it and finish the faded rehearsal beneath it before opening the quiz. The first strategic decision is supplied; the remaining terms, algebra and independent check are yours.
Add the first several terms by hand before using a sum formula. The sequence labs compare a direct construction with the relevant closed form, while the Pascal and binomial labs make their own independent checks visible.
Take the |r| < 1 condition seriously in “Infinite Geometric Series”. The formula returns a number for r = 2 as happily as for r = 1/2, and that number is meaningless.
Build Pascal's triangle by hand once, at least to row 6. “The Binomial Theorem” is much easier for anyone who has seen the coefficients appear before being given a formula for computing one directly.
What it does not cover
Convergence tests for series in general. Only the geometric case is settled here, and it is settled completely; the rest belongs to calculus.
Sequences of functions, power series and Taylor series.
Mathematical induction as a proof technique. Several formulas on this course are proved by other means and stated as proved by induction elsewhere; the Discrete Mathematics path devotes a course to it.