Math Core

Sequences show up everywhere on the AMC: arithmetic and geometric sequences on the AMC 10, and infinite series, telescoping sums and recursions on the AMC 12. The good news is that a small toolkit (two formulas, one pairing trick and one cancellation trick) handles most of them.

Arithmetic sequences

An arithmetic sequence adds the same common difference $d$ each time: $a_n = a_1 + (n - 1)d$ . To add up the terms, pair the first with the last, the second with the second-to-last, and so on. Every pair has the same sum.

Arithmetic and geometric sums

Arithmetic: $a_1 + a_2 + \dots + a_n = \dfrac{n(a_1 + a_n)}{2}$ , the number of terms times the average of the first and last.

Geometric (ratio $r \ne 1$ ): $a_1 + a_1 r + \dots + a_1 r^{n-1} = a_1 \cdot \dfrac{r^n - 1}{r - 1}$ .

Infinite geometric ( $|r| \lt 1$ ): $a_1 + a_1 r + a_1 r^2 + \dots = \dfrac{a_1}{1 - r}$ .

Worked example: Two terms determine the sequence

The $5$ th term of an arithmetic sequence is $17$ and the $12$ th term is $45$ . Find the sum of the first $20$ terms.

Going from term $5$ to term $12$ is $7$ steps, so $7d = 28$ and $d = 4$ . Then $a_1 = 17 - 4 \cdot 4 = 1$ and $a_{20} = 1 + 19 \cdot 4 = 77$ . The sum is $\dfrac{20(1 + 77)}{2} = 780$ .

Geometric sequences

A geometric sequence multiplies by the same ratio $r$ each time: $a_n = a_1 r^{n-1}$ . Dividing two terms eliminates $a_1$ and isolates a power of $r$ .

Worked example: Finding the ratio

A geometric sequence has $a_3 = 12$ and $a_6 = 96$ . Find the sum of the first $8$ terms.

$\dfrac{a_6}{a_3} = r^3 = 8$ , so $r = 2$ and $a_1 = \dfrac{12}{4} = 3$ . The sum is $3 \cdot \dfrac{2^8 - 1}{2 - 1} = 3 \cdot 255 = 765$ .

Telescoping

A sum telescopes when each term can be written as a difference $f(k) - f(k + 1)$ . Then almost everything cancels, leaving only the first and last pieces. The standard ways to create the difference:

Partial fractions: $\dfrac{1}{k(k + 1)} = \dfrac{1}{k} - \dfrac{1}{k + 1}$ .
Rationalizing: $\dfrac{1}{\sqrt{k} + \sqrt{k + 1}} = \sqrt{k + 1} - \sqrt{k}$ .
For products, a ratio $\dfrac{f(k + 1)}{f(k)}$ telescopes the same way.

Worked example: A radical telescope

Evaluate $\displaystyle\sum_{k=1}^{99} \frac{1}{\sqrt{k} + \sqrt{k + 1}}$ .

Multiply each term by $\dfrac{\sqrt{k + 1} - \sqrt{k}}{\sqrt{k + 1} - \sqrt{k}}$ . The denominator becomes $(k + 1) - k = 1$ , so each term is $\sqrt{k + 1} - \sqrt{k}$ . The sum is

(\sqrt{2} - \sqrt{1}) + (\sqrt{3} - \sqrt{2}) + \dots + (\sqrt{100} - \sqrt{99}) = \sqrt{100} - 1 = 9.

Series that are almost geometric

A series like $1 + 2x + 3x^2 + 4x^3 + \dots$ has coefficients growing arithmetically. Multiply by $x$ and subtract: the result is geometric.

Worked example: Shift and subtract

Evaluate $S = 1 + \dfrac{2}{3} + \dfrac{3}{9} + \dfrac{4}{27} + \dots$ .

\begin{aligned} S &= 1 + \tfrac{2}{3} + \tfrac{3}{9} + \tfrac{4}{27} + \cdots \\ \tfrac{1}{3}S &= \phantom{1 + {}} \tfrac{1}{3} + \tfrac{2}{9} + \tfrac{3}{27} + \cdots \end{aligned}

Subtracting, $\dfrac{2}{3}S = 1 + \dfrac{1}{3} + \dfrac{1}{9} + \dots = \dfrac{1}{1 - 1/3} = \dfrac{3}{2}$ . So $S = \dfrac{9}{4}$ .

In general, $1 + 2x + 3x^2 + \dots = \dfrac{1}{(1 - x)^2}$ for $|x| \lt 1$ .

Common mistake

An infinite geometric series only has a sum when $|r| \lt 1$ . If a problem gives a sum and you solve for $r$ , discard any value with $|r| \ge 1$ . Also watch for off-by-one errors: the sequence $7, 11, \dots, 403$ has $\dfrac{403 - 7}{4} + 1 = 100$ terms, not $99$ .

Tip

For arithmetic sequences, the sums of consecutive blocks of the same length ( $S_{10}$ , $S_{20} - S_{10}$ , $S_{30} - S_{20}$ , …) themselves form an arithmetic sequence. That can save a system of equations.

Practice

Practice 1

What is the sum $7 + 11 + 15 + \dots + 403$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

A geometric sequence of positive numbers has second term $6$ and fourth term $54$ . What is the sum of its first five terms?

Practice 3

An infinite geometric series has first term $4$ and sum $12$ . What is the sum of the squares of its terms?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

What is $\left(1 - \dfrac{1}{2^2}\right)\left(1 - \dfrac{1}{3^2}\right)\left(1 - \dfrac{1}{4^2}\right) \cdots \left(1 - \dfrac{1}{50^2}\right)$ ?

Practice 5

A sequence has $a_1 = 5$ and $a_{n+1} = a_n + 2n$ for $n \ge 1$ . What is $a_{50}$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

The sum of the first $10$ terms of an arithmetic sequence is $100$ , and the sum of the first $20$ terms is $300$ . What is the sum of the first $30$ terms?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

For which real number $x$ with $|x| \lt 1$ is $1 + 2x + 3x^2 + 4x^3 + \dots = 4$ ?

Practice 8

What is the value of $\displaystyle\sum_{n=1}^{\infty} \frac{n^2}{2^n} = \frac{1}{2} + \frac{4}{4} + \frac{9}{8} + \frac{16}{16} + \cdots$ ?

Real contest practice

2022 AMC 10A, Problem 20: adding an arithmetic sequence to a geometric one.
2020 AMC 10A, Problem 21: a quotient of powers of $2$ written as a geometric series.
2016 AMC 12B, Problem 14: the infinite geometric series formula, combined with optimization.