Arithmetic and Geometric Sequences

Arithmetic sequences (A.S.)

Each term is the previous term plus a fixed common difference $d$.

$$T(n) = a + (n - 1)d$$ $$S(n) = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}(a + l)$$

where $a$ is the first term and $l$ is the last term.

Geometric sequences (G.S.)

Each term is the previous term times a fixed common ratio $r$.

$$T(n) = ar^{n-1}$$ $$S(n) = \frac{a(r^n - 1)}{r - 1} \quad (r \neq 1)$$

Sum to infinity

If $-1 < r < 1$, the terms get smaller and the sum approaches a limit:

$$S(\infty) = \frac{a}{1 - r}$$

Common mistake: the sum to infinity only exists when $|r| < 1$. For $2, 6, 18, \ldots$ ($r = 3$) there is no sum to infinity.

Finding $a$, $d$ or $r$

Two given terms give two equations. If $T(3) = 10$ and $T(7) = 22$ in an A.S., then $(a + 6d) - (a + 2d) = 12$, so $d = 3$.

In a G.S., divide the two equations instead: $\dfrac{T(5)}{T(2)} = r^3$.

Word problems

• Bouncing ball: a ball dropped from height $h$ that rebounds to a fraction $r$ of its height travels $h + 2 \times \dfrac{hr}{1 - r}$ in total (down once, then up and down for each bounce).
• Least number of terms: set up $S(n) > \text{target}$, solve the inequality, then round up.

Exam tip: state clearly whether the sequence is an A.S. or a G.S. and give $a$ and $d$ (or $r$) before using a formula.