Functions, sequences and custom operations

A function is a rule for substitution

$f(x) = 2x^2 - 3$ is an instruction: whatever goes in the brackets replaces every $x$. So $f(3) = 2(9) - 3 = 15$, and $f(a + 1) = 2(a + 1)^2 - 3$. When the input is an expression, put it in brackets before you substitute; most errors here come from writing $2a + 1^2$ instead of $2(a + 1)^2$.

Function questions in the problem-solving format usually ask one of four things: evaluate at a number, evaluate at an expression, solve $f(x) = k$ for $x$, or compare $f$ at two inputs. All four are substitution followed by ordinary algebra.

Composition: inside first

$f(g(x))$ means: apply $g$ first, then apply $f$ to the result. Work from the inside out. With $f(x) = 2x^2 - 3$ and $g(x) = x + 4$:

$$f(g(-1)) = f(3) = 15, \qquad g(f(-1)) = g(-1) = 3$$

The two orders give different answers, which is normal. Composition is not commutative, so read the brackets carefully.

Custom operations: the question defines the rules

Some questions invent a symbol, for instance "for all numbers $a$ and $b$, let $a \star b = ab - a + b$". The symbol has no meaning beyond that definition. Treat it as a function of two inputs: the first number replaces $a$, the second replaces $b$.

Three habits help:

• Copy the definition and substitute slowly. $5 \star 2 = 10 - 5 + 2 = 7$, while $2 \star 5 = 10 - 2 + 5 = 13$. Order matters unless you can show it does not.
• Nested symbols go inside out, like composition. In $3 \star (2 \star 5)$, compute $2 \star 5$ first.
• Questions about properties ("for which $x$ does $x \star 3 = 3 \star x$?") become equations once you write both sides from the definition.

Arithmetic sequences

In an arithmetic sequence each term adds the same common difference $d$:

$$a_n = a_1 + (n - 1)d$$

The $(n - 1)$ is the point people miss: from term 1 to term $n$ there are $n - 1$ steps. Two known terms give $d$ directly: if $a_4 = 17$ and $a_{10} = 41$, six steps cover $24$, so $d = 4$.

The sum of the first $n$ terms is the number of terms times the average of the first and last:

$$S_n = \frac{n(a_1 + a_n)}{2}$$

That formula works because the terms pair up from the outside in, each pair adding to $a_1 + a_n$. It also gives sums of runs of consecutive integers: $1 + 2 + \dots + 50 = \frac{50 \times 51}{2} = 1275$.

Geometric sequences

Each term multiplies by a common ratio $r$: $a_n = a_1 r^{\,n-1}$. A sequence 3, 6, 12, 24, … has $r = 2$, and its 8th term is $3 \cdot 2^7 = 384$. To find $r$ from two terms, divide them: if $a_2 = 6$ and $a_5 = 162$, then $r^3 = 27$ and $r = 3$.

Recursive definitions

A recursive rule defines each term from the previous one, such as $a_1 = 3$ and $a_{n+1} = 3a_n - 4$. There is usually no shortcut: write the terms one by one ($3, 5, 11, 29, \dots$) and look for a pattern before computing many.

Patterns that repeat

Units digits of powers repeat in short cycles. The units digits of $7^1, 7^2, 7^3, 7^4$ are $7, 9, 3, 1$, and then the cycle starts again. To find the units digit of $7^{43}$, divide the exponent by the cycle length 4: $43 = 4 \times 10 + 3$, so $7^{43}$ ends like $7^3$, in 3. The same idea, list until it repeats then use the remainder, settles many questions about the 100th term of a repeating pattern.