Special sums, differences and modelling
Build the sum of squares, add a series by the method of differences, find the nth term of a quadratic sequence and solve a real problem with an arithmetic series.
- UK: Year 12
- Iran: grade 12
Guide
Sum of squares: Σk2 = n(n + 1)2n + 16 for k from 1 to n. The sum of the integers is nn + 12 and the sum of the cubes is its square. These formulas shorten long sums.
Method of differences: if every term can be written as a "difference of two things", the middle terms cancel. For example 1k(k + 1) = 1k − 1k + 1.
In a quadratic sequence the first differences are not constant but the second differences are. The nth term is an2 + bn + c and the second difference is 2a.
Modelling: first decide if the problem is arithmetic or geometric, then find a, d or r, and n, and put them in the formula.
Common slips: mixing up the sum of squares with the square of the sum; leaving terms uncancelled in the method of differences; taking the first difference as the nth term; using n instead of n − 1 in an arithmetic sum.