Quadratic and Surd Sequences

A sequence lists numbers in a particular order. To investigate how its terms change, subtract each term from the next. These results are the first differences.

Consider

$6,\ 10,\ 16,\ 24,\ 34,\ldots$

Its first differences are

$4,\ 6,\ 8,\ 10.$

They are not constant, so the sequence is not linear. Now find the differences between these differences:

$2,\ 2,\ 2.$

These are the second differences. Because they are constant, the original sequence is quadratic.

A quadratic sequence has a position-to-term rule of the form

$an^2+bn+c,$

where $a\ne0$. You do not need to know that rule before recognising the sequence: for terms at consecutive integer positions, constant second differences are the key sign. This happens because the first differences of a quadratic expression form a linear sequence.

Building and continuing a difference table

Arrange the terms and differences in rows so that each difference lies between the two numbers used to obtain it:

To find the next term, first continue the constant second differences. The next first difference is therefore $10+2=12$. Add this to the last original term: $34+12=46$.

This upward process is important. A quadratic sequence does not usually have one constant number added to every term. Instead, the amount being added changes by a constant amount.

The constant second difference may be negative. For example,

$10,\ 17,\ 22,\ 25,\ 26,\ldots$

has first differences $7,5,3,1$ and second differences $-2,-2,-2$. It is still quadratic; its terms eventually begin to decrease because the first differences keep falling.

Distinguishing quadratic sequences from other patterns

Checking the level at which the differences become constant helps classify a sequence:

• constant first differences indicate a linear sequence;
• constant second differences indicate a quadratic sequence;
• multiplying by the same value each time indicates a geometric sequence.

For example, $3,6,12,24,\ldots$ is geometric, not quadratic: each term is multiplied by $2$. Its first and second differences are not constant.

A few matching differences are useful evidence, but always check all the terms supplied. A pattern can be designed to agree briefly and then change, so the question's stated rule or context also matters.

Surds as exact numbers

A surd is an irrational root written exactly, such as $\sqrt2$. Replacing it with $1.414\ldots$ loses exactness. In sequence work, keeping surds exact makes the pattern easier to see.

Useful facts include

$\sqrt a\times\sqrt b=\sqrt{ab}$

for non-negative $a$ and $b$, and

$(\sqrt a)^2=a.$

Surds should also be simplified where possible. For instance,

$\sqrt8=\sqrt{4\times2}=2\sqrt2.$

Like surds can be collected just like like algebraic terms, so $3\sqrt5+2\sqrt5=5\sqrt5$. Unlike surds generally cannot be combined by addition.

Geometric sequences with a surd ratio

At Higher tier, the common ratio of a geometric sequence may itself be a surd. Consider

$3,\ 3\sqrt2,\ 6,\ 6\sqrt2,\ 12,\ldots$

Each term is obtained by multiplying the previous term by $\sqrt2$:

• $3\times\sqrt2=3\sqrt2$;
• $3\sqrt2\times\sqrt2=3(\sqrt2)^2=6$;
• $6\times\sqrt2=6\sqrt2$;
• $6\sqrt2\times\sqrt2=12$.

The sequence alternates between rational terms and terms containing $\sqrt2$ because multiplying by $\sqrt2$ twice is the same as multiplying by $2$.

You can confirm the common ratio by dividing consecutive terms:

$r=\frac{a_{n+1}}{a_n}.$

For example, $\frac{3\sqrt2}{3}=\sqrt2$. Simplify the quotient exactly before deciding whether the ratio is constant.

To work backwards through a geometric sequence, divide by the common ratio. For the sequence above, the term before $3$ is

$3\div\sqrt2=\frac{3}{\sqrt2}=\frac{3\sqrt2}{2}.$

When answering a sequence question involving surds, identify the operation, simplify after every step and check that the same exact relationship holds throughout.