Reduction

In one sentence

Reduction: transform a problem you cannot solve, step by step, into a problem you already know how to solve. As the old joke goes: how do you boil water in a kettle that is already full? Pour the water out. Now it is the “empty kettle” problem, and you solved that long ago.

The four most common routes: complicated to simple, unfamiliar to familiar, general to special, and one statement to an easier equivalent one. The picture above is “unfamiliar to familiar”: nobody has seen $a_{n + 1} = 3 a_{n} + 4$ before, but everyone knows geometric sequences.

Example: a₁ = 1, aₙ₊₁ = 3aₙ + 4. Find the general term

Step 1 / 4

An unfamiliar recurrence
Compute a few terms: $1, 7, 25, 79, \dots$ . The ratios of consecutive terms, $7, 3.57 \dots, 3.16 \dots$ , are not constant: this is not a geometric sequence.

$a_{n + 1} = 3 a_{n} + 4$
Make it look familiar
Find c
Reduction complete

Predict first

Take the recurrence $a_{n + 1} = 5 a_{n} - 8$ . For $b_{n} = a_{n} + c$ to be a geometric sequence, what should $c$ be?

c = - 2

c = 2

c = - 8

c = 8

Choose an answer first; then you can see the result.

Stop 1 · Substitution in functional equations (Vol. 2)

When $f (x)$ and $f (\frac{1}{x})$ are tangled together, one equation is not enough. Put $\frac{1}{x}$ in place of $x$ to create a second equation, and the functional equation has been reduced to a system of equations.

Example: for x ≠ 0, f(x) + 2f(1/x) = 3x. Find f(x)

Step 1 / 4

One equation, two unknowns
Think of $f (x)$ and $f (\frac{1}{x})$ as two unknowns $A$ and $B$ . With only one equation, you cannot solve for them.

$A + 2 B = 3 x$
Replace x by 1/x
Reduced to a system
Substitute back and check

Stop 2 · Trigonometric substitution (Vol. 3)

$1 - x^{2}$ is a headache, but when $x = cos t$ it is simply $sin t$ . A problem about radicals is reduced to a problem about trigonometry, and we know how to find the maximum and minimum of a trigonometric function.

One problem, several solutions

Find the range of the function $y = x + 1 - x^{2}$ .

Trigonometric substitution

The domain is $[- 1, 1]$ . Let $x = cos t$ with $t \in [0, π]$ . Then $1 - x^{2} = sin t \geq 0$ .

$y = cos t + sin t = 2 sin (t + \frac{π}{4})$ , with $t + \frac{π}{4} \in [\frac{π}{4}, \frac{5 π}{4}]$ .

So $sin (t + \frac{π}{4}) \in [- \frac{2}{2}, 1]$ and $y \in [- 1, 2]$ .

CostOne reduction, but you must state the range of the new variable.

Isolate the root and square

$y - x = 1 - x^{2}$ requires $y \geq x$ . Squaring gives $2 x^{2} - 2 y x + y^{2} - 1 = 0$ , and discriminant $\geq 0$ gives $y^{2} \leq 2$ .

Only after checking the constraint $y \geq x$ and the endpoints $x = \pm 1$ do you get $y \in [- 1, 2]$ . That is one more layer of case analysis than the substitution.

CostIt works, but watch out for extraneous roots and signs introduced by squaring.

Stop 3 · Translating the axes (Vol. 7)

What curve does $x y - 2 x + 3 y = 0$ describe? It looks unfamiliar. Turn it into a product, translate the axes, and it is reduced to a familiar rectangular hyperbola $x^{'} y^{'} = k$ .

Example: simplify xy − 2x + 3y = 0

Step 1 / 4

Make a product
$(x + 3) (y - 2) = x y - 2 x + 3 y - 6$ , which is $6$ less than the left-hand side of the equation.

$(x + 3) (y - 2) = x y - 2 x + 3 y - 6$
Rewrite the equation
Translate the axes
Now you recognize it

Stop 4 · Equations with complex numbers (Vol. 4)

Expanding $z^{3} = 8 i$ in algebraic form $z = a + bi$ leaves you with a system of equations of degree three. Switch to trigonometric form, and solving the equation is reduced to “take the root of the modulus, divide the argument by three”.

Example: solve z³ = 8i

Step 1 / 4

Write it in trigonometric form
$8 i$ has modulus $8$ and argument $\frac{π}{2}$ .

$8 i = 8 (cos \frac{π}{2} + i sin \frac{π}{2})$
Root of the modulus, argument divided by three
Three roots
Check one

Practice

Exercise 1Basic

Given $a_{1} = 1$ and $a_{n + 1} = 2 a_{n} + 1$ , find $a_{10}$ .

Exercise 2Basic

A function $f (x)$ satisfies $f (x) + 2 f (1 - x) = x$ for every real $x$ . Find $f (2)$ .

Exercise 3Advanced

Find the maximum of $y = 3 x + 4 1 - x^{2}$ .

Exercise 4Challenge

Given $a_{1} = 1$ and $a_{n + 1} = \frac{a _{n}}{1 + a _{n}}$ , find $a_{20}$ .

In one sentence

Example: a₁ = 1, aₙ₊₁ = 3aₙ + 4. Find the general term

Stop 1 · Substitution in functional equations (Vol. 2)

Example: for x ≠ 0, f(x) + 2f(1/x) = 3x. Find f(x)

Stop 2 · Trigonometric substitution (Vol. 3)

Trigonometric substitution

Isolate the root and square

Stop 3 · Translating the axes (Vol. 7)

Example: simplify xy − 2x + 3y = 0

Stop 4 · Equations with complex numbers (Vol. 4)

Example: solve z³ = 8i

Practice

Mistake Museum · Exhibits from this stop

Substituting, then forgetting the range of the new variable

Substituting only once and hoping to find f(x)

Stops on this line