Arithmetic and geometric sequences and series

Q: What is the difference between a sequence and a series?

A sequence is a list of numbers in order, like 2, 5, 8, 11. A series is what you get when you add those numbers: 2 + 5 + 8 + 11 = 26. Sequence questions ask for a term; series questions ask for a sum.

Q: How do I tell if a sequence is arithmetic or geometric?

Subtract neighbors: if the differences are all equal, it is arithmetic. Divide neighbors: if the ratios are all equal, it is geometric. If neither is constant, it is some other kind of sequence, and these formulas do not apply.

Q: When does an infinite geometric series have a sum?

Only when the ratio is between -1 and 1, so |r| < 1. Then the terms shrink toward 0 fast enough for the total to settle at \frac{a_1}{1 - r}. If |r| \ge 1, the terms do not shrink, and the sum has no finite value.

Q: What does sigma notation mean?

The symbol \sum means "add up". \sum_{k=1}^{4} 2k means put k = 1, 2, 3, 4 into 2k and add: 2 + 4 + 6 + 8 = 20. The numbers below and above the sigma tell you where k starts and stops.

The key idea

Look at how you get from one term to the next. If you add the same amount every time, the sequence is arithmetic. If you multiply by the same amount, it is geometric.

The four formulas
	Arithmetic (add d)	Geometric (multiply by r)
Example	$4, 7, 10, 13, \ldots$ with $d = 3$	$4, 12, 36, 108, \ldots$ with $r = 3$
nth term	$a_n = a_1 + (n - 1)d$	$a_n = a_1 r^{n - 1}$
Sum of n terms	$S_n = \frac{n}{2}(a_1 + a_n)$	$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$
Sum of all terms	No finite sum (unless every term is 0)	$S = \frac{a_1}{1 - r}$ if $\|r\| < 1$

The $n - 1$ is there because the first term has had no steps yet. The 40th term is 39 steps after the first.

S_\infty = \frac{a_1}{1 - r}, \quad |r| < 1

Worked examples

Example 1: a term far down an arithmetic sequence

Problem Find the 40th term of $7, 11, 15, 19, \ldots$

The common difference is the gap between terms.
$11 - 7 = 4$
Use $a_n = a_1 + (n - 1)d$ with $a_1 = 7$ , $n = 40$ , $d = 4$ .
$7 + (40 - 1)(4) = 7 + 156 = 163$

Answer $a_{40} = 163$

Example 2: an arithmetic series

Problem Find the sum $3 + 8 + 13 + \cdots + 98$ .

The difference is 5. First count the terms: how many steps of 5 from 3 to 98, plus 1 for the first term.
$n = \frac{98 - 3}{5} + 1 = 20$
Average of the first and last term, times the number of terms.
$\frac{20}{2}(3 + 98) = 10 \cdot 101 = 1010$

Answer The sum is 1,010.

Example 3: a geometric term and sum

Problem For $5, 15, 45, \ldots$ , find the 8th term and the sum of the first 8 terms.

The ratio is $15 \div 5 = 3$ . Use $a_n = a_1 r^{n - 1}$ .
$5 \cdot 3^{8 - 1} = 5 \cdot 2187 = 10935$
Use $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ with $n = 8$ .
$5 \cdot \frac{1 - 3^8}{1 - 3} = 5 \cdot \frac{-6560}{-2} = 16400$

Answer The 8th term is 10,935 and the sum of the first 8 terms is 16,400.

Example 4 (test-hard): a repeating decimal as a fraction

Problem Write $0.\overline{27} = 0.272727\ldots$ as a fraction using a geometric series.

Split it into blocks: $0.27 + 0.0027 + 0.000027 + \cdots$ . That is geometric with $a_1 = \frac{27}{100}$ and $r = \frac{1}{100}$ .
Since $|r| < 1$ , use the infinite sum formula.
$\frac{\frac{27}{100}}{1 - \frac{1}{100}} = \frac{27}{99} = \frac{3}{11}$
Check: $3 \div 11 = 0.272727\ldots$

Answer $\frac{3}{11}$

Common mistakes

Using n instead of n - 1. The 40th term of $7, 11, 15, \ldots$ is $7 + 39 \cdot 4$ , not $7 + 40 \cdot 4$ . Fix: count steps between terms, not terms.
Miscounting the number of terms. From 3 to 98 by 5s is 20 terms, not 19. Fix: $n = \frac{\text{last} - \text{first}}{d} + 1$ .
Calling a sequence geometric because it grows fast. Check the ratios: $2, 6, 12, 20$ has ratios 3, 2, $\frac{5}{3}$ , so it is neither type.
Using the infinite sum formula when $|r| \ge 1$ . $2 + 4 + 8 + \cdots$ has no finite sum. Fix: check that r is between $-1$ and 1 first.
Getting the ratio upside down. For $81, -27, 9, \ldots$ , r is $\frac{-27}{81} = -\frac{1}{3}$ . Fix: divide a term by the one before it.

Quick methods

Practice

5 practice questions

What is the 10th term of the arithmetic sequence $50, 46, 42, \ldots$ ?
1. $10$
2. $14$
3. $18$
4. $86$
Show answer
Answer: $14$
$d = -4$ , so $a_{10} = 50 + 9(-4) = 14$ . 10 uses 10 steps instead of 9, and 86 adds 4 each time instead of subtracting.
What is the common ratio of the geometric sequence $81, -27, 9, \ldots$ ?
1. $-3$
2. $-\frac{1}{3}$
3. $\frac{1}{3}$
4. $3$
Show answer
Answer: $-\frac{1}{3}$
Divide a term by the one before it: $\frac{-27}{81} = -\frac{1}{3}$ . The signs alternate, so r is negative, and the terms shrink, so $|r| < 1$ .
What is the sum of the first 30 positive odd numbers, $1 + 3 + 5 + \cdots + 59$ ?
1. $450$
2. $870$
3. $900$
4. $930$
Show answer
Answer: $900$
There are 30 terms with average $\frac{1 + 59}{2} = 30$ , so the sum is $30 \cdot 30 = 900$ . In general, the first n odd numbers add to $n^2$ .
What is the sum of the infinite series $9 + 6 + 4 + \cdots$ ?
1. $13.5$
2. $19$
3. $27$
4. The series has no finite sum.
Show answer
Answer: $27$
$r = \frac{6}{9} = \frac{2}{3}$ , which is less than 1, so the sum is $\frac{9}{1 - \frac{2}{3}} = 27$ . 13.5 uses $r = \frac{1}{3}$ , and 19 adds only the three terms shown.
In an arithmetic sequence, $a_5 = 17$ and $a_{12} = 45$ . What is $a_1$ ?
Show answer
Answer: 1
$d = \frac{45 - 17}{12 - 5} = 4$ . Going back 4 steps from $a_5$ : $a_1 = 17 - 4 \cdot 4 = 1$ .

Frequently asked questions

What is the difference between a sequence and a series?

A sequence is a list of numbers in order, like $2, 5, 8, 11$ . A series is what you get when you add those numbers: $2 + 5 + 8 + 11 = 26$ . Sequence questions ask for a term; series questions ask for a sum.

How do I tell if a sequence is arithmetic or geometric?

Subtract neighbors: if the differences are all equal, it is arithmetic. Divide neighbors: if the ratios are all equal, it is geometric. If neither is constant, it is some other kind of sequence, and these formulas do not apply.

When does an infinite geometric series have a sum?

Only when the ratio is between $-1$ and 1, so $|r| < 1$ . Then the terms shrink toward 0 fast enough for the total to settle at $\frac{a_1}{1 - r}$ . If $|r| \ge 1$ , the terms do not shrink, and the sum has no finite value.

What does sigma notation mean?

The symbol $\sum$ means "add up". $\sum_{k=1}^{4} 2k$ means put $k = 1, 2, 3, 4$ into $2k$ and add: $2 + 4 + 6 + 8 = 20$ . The numbers below and above the sigma tell you where k starts and stops.

Sequences and series

The key idea

Worked examples

Common mistakes

Quick methods

Practice

Frequently asked questions

Try asking Ducky

The key idea

Worked examples

Common mistakes

Quick methods

Practice

Frequently asked questions

Related

Try asking Ducky