A sequence is a collection of numbers (called terms) connected by some definite mathematical law. Thus the numbers 1, 2, 4, 8, 16 form a sequence where the law is that the first term is 1 and every other term is double the preceding term. This is a finite sequence if we stop at 16. However if we let the terms continue doubling forever then we would have an infinite sequence. The same law of doubling governs the sequences 5, 10, 20, 40, 80 and 12, 24, 48, 96, 192. These sequences are different simply because they start with a different number. In general we use the letter $a$ to represent the terms of a sequence. The first term is denoted by $a_1$, the second term by $a_2$, the third term by $a_3$ and so on. If a sequence is governed by the law of doubling then:

the first term is $a_1$

the second term is $a_2 = 2\times a_1 = 2^1a_1$

the third term is $a_3 = 2\times a_2 = 2\times2^1a_1 = 2^2a_1$

the fourth term is $a_4 = 2\times a_3 = 2\times 2^2 a_1 = 2^3 a_1$

et cetera and in general the $n$-th term is $a_n = 2^{n-1}a_1$

Exercises:

1. Calculate the twentieth term of the sequence 5, 10, 20, 40, 80 . . .

2. The law governing the sequence 2, 6, 12, 20, 30 . . .  has two equivalent forms. What are they? Show that $a_n = n(n + 1)$ and calculate the twentieth term.

3. What law governs the sequence 6, 7, 9, 12, 16 . . . ? Show that $a_n = a_{n-1} + n – 1$ and calculate the tenth term.

4. What law governs the sequence 6, 11, 20, 36, 107, 177, 286 . . . ? Find the eight term.