Sequences — Objectives | Mathematics Department

Representing Sequences

1. Represent a sequence using pictures, tables of values, graphs and algebra.
2. Describe a sequence using a term-to-term rule (in words) and a position-to-term rule (using $n$).
3. Identify whether a sequence is arithmetic, geometric, quadratic or neither.
4. Use the general rule of a sequence to predict any $n$th term.
5. Test whether a given number is a term of a sequence by solving an equation.

6. Determine the general rule $u_n = u_1 + (n-1)d$ for an arithmetic sequence.
7. Determine the general rule $u_n = u_1 \cdot r^{n-1}$ for a geometric sequence.
8. Work backwards from given terms to find $u_1$, $d$, $r$ or $n$ (e.g. given $u_5 = 2$ and $u_8 = 8$, find $r$).
9. Use properties of sequences to solve for unknowns (e.g. consecutive arithmetic terms $k$, $2k+2$, $4k+4$; consecutive geometric terms $2$, $k$, $10$).
10. Write a recursive rule for an arithmetic sequence using $u_{n+1} = u_n + d$.
11. Write a recursive rule for a geometric sequence using $u_{n+1} = u_n \cdot r$.

12. Determine the general rule from a shifted square or cube sequence (e.g. one less than the square numbers gives $u_n = n^2 - 1$).
13. Calculate the sum of the first $n$ terms of an arithmetic series.
14. Calculate the sum of the first $n$ terms of a geometric series.
15. Discuss the behaviour of an infinite geometric series and identify when it converges.