Sequences and figurate numbers

Arithmetic sequences

A sequence is an ordered list of numbers, and it is arithmetic when each term changes by the same fixed amount — the common difference $d$. Find $d$ by subtracting any term from the one after it. It can be negative ($19, 15, 11, \dots$ has $d = -4$) or even zero. The powers of 2 and the Fibonacci numbers are not arithmetic: their differences keep changing.

Terms are named by position: $t_1$ is the first term, and $t_n$ is the $n$th. In a statement like $t_{n+1} = t_n + 5$, the $+1$ sits in the subscript so it is about position; the $+5$ is about value — “the next term is 5 more than the current one”.

The nth term

With first term $a$ and common difference $d$:

$$t_n = a + (n - 1)d$$

Two known terms pin down the sequence. If $t_4 = 15$ and $t_{10} = 33$, the six steps between them cover $33 - 15 = 18$, so $d = 3$, and working back gives $a = 6$.

To test whether a number belongs to a sequence, set $t_n$ equal to it and solve for $n$. Is 90 a term of $7, 12, 17, \dots$? Solving $7 + (n-1) \times 5 = 90$ gives $n - 1 = 16.6$, which is not a whole number — so no.

Arithmetic series

A series is what you get by adding the terms of a sequence. Young Gauss famously summed 1 to 100 by pairing opposite ends — $1 + 100$, $2 + 99$, and so on — giving 50 pairs of 101, or 5050. The same pairing idea gives the general formulas, where $\ell$ is the last term being added:

$$S_n = \frac{n}{2}(a + \ell) = \frac{n}{2}\left(2a + (n-1)d\right)$$

Example: $5 + 10 + 15 + \dots + 100$ has 20 terms, so $S_{20} = \frac{20}{2}(5 + 100) = 1050$.

Figurate numbers

Adding more and more terms of simple arithmetic sequences builds the figurate numbers — numbers you can draw as dot patterns:

• Triangular: $T_n = 1 + 2 + \dots + n = \frac{n(n+1)}{2}$
• Square: $Q_n = 1 + 3 + 5 + \dots + (2n-1) = n^2$ — the sum of the first $n$ odd numbers
• Pentagonal: $P_n = 1 + 4 + 7 + \dots = \frac{n(3n-1)}{2}$

The underlying common differences are 1, 2 and 3 respectively.

Proving identities

Exams ask you to show facts like $Q_n = T_n + T_{n-1}$ or $Q_n = 2T_n - n$. Start from one side, substitute the formulas, and simplify step by step until it equals the other side — never start by assuming both sides are equal. A dot diagram (splitting a square into two triangles) often earns the “show that” mark alongside the algebra.