Sequences and series
Algebra 2. Drag the sliders, type your own rules, then test yourself.
What is a sequence?
A sequence is an ordered list of numbers, called terms: a₁, a₂, a₃, … A series is what you get when you add the terms. Two ways to describe a sequence:
Explicit gives any term directly. Recursive builds each term from the one before.
Try: 5 − 2n · 2^n · 3*2^(n-1) · n^2
Try: a + 5 · 2a · a/2 · 2a + 1. Notice how 3a − 1 is neither arithmetic nor geometric, and how quickly the bars grow.
Spot the pattern
Subtract neighbors. If the difference is always the same, the sequence is arithmetic. Divide neighbors. If the ratio is always the same, it is geometric. The next two tabs take each in turn, and the Other patterns tab covers sequences that are neither.
Arithmetic sequences
Add the same number, the common difference d, every time.
The bars rise or fall in equal steps. That straight-line growth is what “arithmetic” looks like. Pair the first and last terms, the second and second-to-last, and so on: every pair has the same total, which is why Sₙ works.
Geometric sequences
Multiply by the same number, the common ratio r, every time.
Slide r between −1 and 1. The terms shrink toward 0 and the partial sums settle on one value, so the infinite series converges. Push |r| to 1 or more and the sums run away.
Neither arithmetic nor geometric
Plenty of sequences follow a clear rule without a constant difference or ratio. A reliable move is to take differences of the differences until a row becomes constant. The number of rows it takes tells you how the sequence grows.
Seeing a geometric series converge
Take a square with area 1. Shade half of it, then half of what is left, then half of that again. The shaded areas ½ + ¼ + ⅛ + … form a geometric series. Shade as many pieces as you like: the total gets closer and closer to 1 but never goes past it. Try other fractions with the slider.
Hop along the number line
Each partial sum is a stop on the number line and each new term is a hop. Change a₁ and r and watch where the hops lead. Hops to the right arc above the line, and hops to the left arc below it.
Look for r between −1 and 1 (the hops shrink and the sums settle), then drag r past 1 or below −1 to see them run away.
The frog in the hole
A frog sits at the bottom of a hole. Each day it climbs u meters, and each night it slips back s meters. How many days until it gets out? The answer uses an arithmetic sequence, with one trap.
Teal segments are daytime climbs and orange segments are night slips. Try s = 0, then make s close to u and watch the days pile up.
Fibonacci in the world
The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … show up in real growth patterns, and they also attract a lot of myths. Here are the patterns you can build yourself, then a fact-check.
Shells and hurricanes have spiral shapes too, but the match to this exact spiral is loose. A nautilus shell is a logarithmic spiral with a growth rate that is not the golden ratio.
Nudge the angle by even half a degree and gaps open up. Try 120° or 144°: simple fractions of a full turn make straight rays, wasting space. The golden angle is 360° ÷ φ², and it is the “most irrational” turn, so no seed ever lines up exactly with an earlier one.
The golden spiral is just one member of this family, with a growth factor of 1.618 per quarter-turn. Real shells, including conchs and nautilus, grow at their own rates, roughly 3× per full turn for a nautilus (about ×1.33 per quarter-turn), so they are close cousins of the golden spiral and not copies of it.
Petal numbers vary from flower to flower and many species don’t follow the pattern, so treat these as common counts, not rules. Fibonacci numbers also appear in the spiral scales of pinecones and pineapples (often 8 and 13) and in the swirls of romanesco broccoli, and these make good photos to look for.
Real: architect Le Corbusier built his Modulor measurement system around the golden ratio and Fibonacci-like series. Italian artist Mario Merz put Fibonacci numbers into neon installations. Salvador Dalí is reported to have used golden-ratio proportions in The Sacrament of the Last Supper.
Overstated: claims that the Parthenon, the Pyramids or the Mona Lisa were designed with the golden ratio rest on loose measurements and no firm evidence. If you pick enough rectangles in a picture, one will come close to 1.618.
A good test for any claim: does the pattern come from a growth rule you can state (as with the seeds), or only from drawing a spiral over a picture?
Try to make the spiral’s center land on something important, such as a face. Notice how much flipping, rotating and sliding it takes. If you can make a spiral fit almost any picture by adjusting it, a good fit is weak evidence. Compare this with the sunflower, where the spiral comes from the growth rule.
Step puzzles
Fill every empty circle so the numbers along each line, every row and every column, form an arithmetic sequence. Lines turn green when they work and red when they don’t. This puzzle style comes from Naoki Inaba. The puzzles here are generated fresh each time, and each has exactly one solution.
Strategy: find a line with two known numbers. If they are j steps apart, the common difference is their gap divided by j. Fill in that line, and the new numbers unlock the lines that cross it. Negative differences are allowed.
Sigma notation
Σ (sigma) means “add up.” The number below is where k starts, the number above is where it stops.
Up to 30 terms. Try 3*2^k or k^2 (neither is arithmetic or geometric, and k² is neither).
Practice
Ten fresh problems every time. Pick a topic and a difficulty. Answers can be whole numbers or decimals.