Patterns built by a rule

This lesson asks you to determine pattern rules and use them to extend patterns, to predict and justify predictions, and to find missing elements. The museum offers one rule for building a table of numbers from the row before it; it does not serve growing, shrinking or linear patterns, graphs or algebraic expressions, which your class brings from elsewhere. ¹

Write the rule in words first, then in a table of values. Predict the next two rows, and check your prediction by applying the rule.

Al-Karaji's rule builds each new row of numbers from the row before it. The sources describe it so: the square gives one, two and one ². To get the next list, keep a one at each end, add one and two to get three, and then two and one to get another three ². The page he wrote is not in front of us: the record says that al-Samawal transmits the account of an original computation that no longer survives ^{2 3}.

The records preserve no sentence in which al-Karaji says why he worked out these numbers, and the museum does not supply one. They show where the rule belongs: early-eleventh-century Baghdad algebra ^{2 3}. MacTutor reports that he calculated the cube of a sum in al-Fakhri, and the cube of a difference and fourth power of a sum in al-Badi ². Al-Samawal admired him but also identified shortcomings and developed improvements ^{4 3}.

The table is often called Pascal's triangle. The record says the name does not imply that the triangle was unknown before Pascal, and that MacTutor says so expressly ⁵. The history of Jaina mathematics describes a triangle in a tenth-century commentary on an older work ⁶. For China, MacTutor describes Jia Xian's table ⁷. The record calls this comparable work, not something shown to have been borrowed from al-Karaji ^{7 2}.

The record checks the rule with chosen numbers, but says this does not alone prove the rule for every power ². Why is an example that agrees not enough, and what else would you want to see before you trusted the rule for every row?

• Al-Karaji's coefficient rule: how each row is made from the row before, and what is lost.
• al-Karaji: his works, his name, and what is not known about him.
• Al-Bahir fi al-jabr: the book that preserves the calculation.
• al-Samaw'al al-Maghribi: the later writer who passes it on.

Read the record on al-Karaji's binomial-coefficient recurrence. Build the next rows of the table by adding neighbours, and write your rule as one sentence.