Algebra

After arranging the numbers in increasing or decreasing order:

• If the number of terms is odd, the median is the middle number.
• If the number of terms is even, the median is the average of the middle two numbers.

The harmonic mean of the numbers $a_1, a_2, a_3, \ldots, a_n$ is

This means the harmonic mean is

An arithmetic sequence is a sequence of numbers with the same difference between consecutive terms.

is an arithmetic sequence because there is always a difference of $3$ between consecutive terms.

In general, the terms of an arithmetic sequence can be represented as

where

• $d$ is the common difference between consecutive terms
• $n$ is the number of terms

which means the $n$th term of an arithmetic sequence is

We also have

which means

Essentially,

which essentially means

• If the number of terms is even, the average is the average of the middle $2$ terms.
• If the number of terms is odd, the average is the middle term.

which essentially means

We can also substitute $a_n = a_1 + (n - 1)d$ to get

A geometric sequence is a sequence of numbers with the same ratio between consecutive terms.

is a geometric sequence because there is always a ratio of $2$ between consecutive terms.

which means

A general form for calculating the $n$th term is

which means

For $r \ne 1$,

which essentially means

For $-1 < r < 1$,

In simple terms, the sum of the first $n$ odd numbers is $n^2$.

To think about it intuitively, just take $2$ out of each term:

In simple terms, the sum of the first $n$ even numbers is $2 \times$ the sum of the first $n$ numbers.

Expand the first few and last few terms, and cancel out any terms you see.

Partial fraction decomposition is a telescoping technique in which you split terms into multiple terms so that terms cancel. In general, to find the partial fraction decomposition of $\frac{1}{ab}$ for any expressions $a$ and $b$, we write the equation

and then solve for $x$ and $y$. For example, the partial fraction decomposition of

(in this case $a = n$ and $b = n + 1$) is

Using partial fraction decomposition, we can telescope and evaluate expressions easily.

Equivalently,

Equivalently,

This is one of the most common topics on the AMC 10 especially. The trick to solving these problems is to assign variables to the unknowns in the problem and solve.

Once you've found the system of equations in your word problem (or the problem gives you a system of equations), consider one or more of the following methods for solving it.

1. Substitution

2. Elimination

3. Diagonal Product Method (only for 2-variable equations)

4. Adding/Subtracting Equations

5. Be on the lookout for common factorization tricks (see Section 1.10)

6. Constructing polynomials from your equations using Vieta's Formulas (see Section 1.9)

7. Graphing your equations (we recommend graph paper)

8. Using symmetry

9. Creating a geometric setup using geometric formulas such as the

   • Law of Cosines
   • Heron's Formula
   • Sine area formula
   • Stewart's Theorem
   • Trig Identities

   (See the geometry chapter for these formulas.)

In the quadratic formula,

(the part inside the square root) is the discriminant of the quadratic.

1. If the discriminant $b^2 - 4ac$ is $0$, then the quadratic has a double (repeated) root.
2. If the discriminant $b^2 - 4ac$ is positive, the quadratic has $2$ different real roots.
3. If the discriminant $b^2 - 4ac$ is negative, the quadratic has no real roots.

The solutions to the quadratic equation

are

In a quadratic equation

the sum of its roots is

and the product of its roots is

In a polynomial

with roots $r_1, r_2, r_3, \ldots, r_n$, the following hold:

Note that the signs alternate: a sum of products of an odd number of roots gets a minus sign, and an even number gets a plus sign.

In a polynomial

where $a_n$ is nonzero and every $a_i$ is an integer, all integer roots of the polynomial must divide $a_0$, the constant term of the polynomial.

The remainder when a polynomial $P(x)$ is divided by $x - r$ is $P(r)$.

$x - r$ divides a polynomial $P(x)$ if $P(r) = 0$.

This is a direct consequence of the Remainder Theorem.

A polynomial

can be expressed in the form

where $r_1, r_2, r_3, \ldots, r_n$ are the $n$ roots of the polynomial.

If $a + bi$ (with $a$, $b$ real) is a root of a polynomial with real coefficients, then $a - bi$ is a root too.

If the polynomial

has roots $r_1, r_2, r_3, \ldots, r_n$, then

has roots

Essentially, when you flip the coefficients of a polynomial, its roots become the reciprocals of the original roots.

If the polynomial

has roots $r_1, r_2, r_3, \ldots, r_n$, then

has roots

Let the polynomial

have roots $r_1, r_2, r_3, \ldots, r_n$, and let

Then the following hold:

Note that $a_j = 0$ whenever $j$ is negative, as that might show up in your expansion.

Essentially, this says:

1. Start with $S_k$ and multiply it by the leftmost coefficient $a_n$.
2. Then multiply $S_{k-1}$ by the next coefficient $a_{n-1}$.
3. Continue doing so and summing the products until either
   • you reach $S_0$, in which case instead of multiplying $S_0$ by that last coefficient you multiply it by $k$, or
   • the coefficients run out, in which case you simply add the last term and stop.
4. Set the sum equal to $0$.

If

then

To solve a symmetric polynomial $P(x) = a_0 x^n + a_1 x^{n-1} + \cdots + a_1 x + a_0$ of even degree:

• Divide by $x^{n/2}$.
• Group the $x^k$ and $\frac{1}{x^k}$ terms together.
• Make the substitution

and write all the terms in your expression that way.

• Solve the reduced polynomial.

For $x \ne 0$:

Sum of odd powers:

The signs in the second factor alternate between positive and negative.

Difference of powers:

The signs in the second factor are all positive.

Here are some ideas for algebraic manipulation:

1. Group similar terms together.
2. Be on the lookout for factorizations.
3. Take advantage of symmetry; you can use it to construct polynomials.
4. Make smart substitutions that simplify your expression. For example, if the term

appears multiple times in your expression, just let

to simplify it.

For real $x$,

This means all perfect squares are $0$ or more.

In a quadratic $Q(x) = ax^2 + bx + c$,

• If $a > 0$, then the minimum value of $Q(x)$ is $c - \frac{b^2}{4a}$ and occurs when $x = -\frac{b}{2a}$.
• If $a < 0$, then the maximum value of $Q(x)$ is $c - \frac{b^2}{4a}$ and occurs when $x = -\frac{b}{2a}$.

For non-negative reals $a$ and $b$,

Basically, this means the average of $2$ non-negative numbers (arithmetic mean) is always at least as big as the square root of the product of the $2$ numbers (the geometric mean).

Equality occurs when $a = b$.

For $x > 0$, the minimum value of $x + \frac{1}{x}$ is $2$ and occurs when $x = 1$.

For positive $a$ and $b$:

• The minimum value of $a + b$ (if $ab$ remains constant) occurs when $a = b$.
• The maximum value of $ab$ (if $a + b$ remains constant) occurs when $a = b$.

For non-negative reals $a_1, a_2, \ldots, a_n$,

Equality occurs when $a_1 = a_2 = \cdots = a_n$ (all the variables are equal).

Another way to say this is

For non-negative reals $a_i$, $c_i$,

For reals $a_i$ and $b_i$,

This means

Equality holds when the ratio $\frac{a_i}{b_i}$ is the same for all $i$.

For reals $a_i$ and positive reals $b_i$,

Alternately (for non-negative $a_i$),

Note that this is a direct consequence of Cauchy-Schwarz.