Ximpledu

Sequence

See how to find a sequence and a series
(arithmetic/geometric/other).
23 examples and their solutions.

Arithmetic Sequence

Definition

An arithmetic sequence is a sequence
whose difference of the adjacent terms
are the same.
So, if you add +d,
you get the next term.

Formula

an = a + (n - 1)d

an: nth term
a: first term, a1
d: common difference

Example

1, 4, 7, 10, 13, ...
an = ?
Solution

Example

-2, 5, 12, 19, ...
ak = 551, k = ?
Solution

Example

Arithmetic sequence: a8 = 5, a12 = 13
an = ?
Solution

Arithmetic Sequence: Mean

Definition

Arithmetic Means:
the middle terms that form an arithmetic sequence
with the first and the last term.

Example

Find the three arithmetic means between 7 and 23.
Solution

Arithmetic Series

Formula

Sn = n2[2a + (n - 1)d]
= n2[a + an]

Sn: a1 + a2 + a3 + ... + an
a: first term, a1
d: common difference
Series: partial sum of a sequence, Sn

Example

Arithmetic sequence: a = 3, d = 5
S20 = ?
Solution

Example

-1 + 3 + 7 + 11 + ... + 123
Solution

Example

Sn = n2 + 2n
an = ?
Solution

Sigma (Math)

Definition

k = 1nak = a1 + a2 + a3 + ... + an
Sigma, ∑, is a way
to write the sum of a sequence (series).
k = 1n ak is read as
[the sum of ak as k goes from 1 to n].
Sigma (Math)

Geometric Sequence

Definition

A geometric sequence is a sequence
whose ratio of the adjacent terms
are the same.
So, if you multiply ×r,
you get the next term.

Formula

an = arn - 1

an: nth term
a: first term, a1
r: common ratio

Example

2, 6, 18, 54, 162, ...
an = ?
Solution

Example

320, 160, 80, 40, ...
ak = 58, k = ?
Solution

Example

Geometric sequence: a2 = -6, a5 = 48
an = ?
Solution

Geometric Sequence: Mean

Definition

Geometric Means:
the middle terms that form a geometric sequence
with the first and the last term.

Example

Find the four geometric means between 6 and 192.
Solution

Example

Find the three geometric means between 5 and 405.
Solution

Geometric Series

Formula

Sn = a(rn - 1)r - 1
= a(1 - rn)1 - r

Sn: a1 + a2 + a3 + ... + an
a: first term, a1
r: common ratio

Example

Geometric sequence: a = 3, r = 2
S5 = ?
Solution

Example

Geometric sequence: a1 = 4, a3 = 36
S5 = ?
Solution

Example

k = 14 5⋅(23)k
Solution

Infinite Geometric Series

Infinite Geometric Sequence

Formula

S = a1 - r (-1 < r < 1)

S: a1 + a2 + a3 + ...
a: first term, a1
r: common ratio

Example

1 + 12 + 14 + 18 + ...
Solution

Example

10 - 203 + 409 - 8027 + ...
Solution

Example

n = 0 8⋅(17)n
Solution

Repeating Decimal

Definition

0.123 = 0.1232323...
A repeating decimal is a decimal
that has a repeating part.
The numbers under the bar is the repeating part.
A repeating decimal is a rational number.
So you can change a repeating decimal
to a fraction.

Example

0.123 → fraction?
Solution

Recursive Formula

Example

a1 = 4, an + 1 = an + 6
a1 ~ a4 = ?
Solution

Example

a1 = -2, an + 1 = an + 3n
a1 ~ a4 = ?
Solution

Example

a1 = 1, a2 = 1, an + 2 = an + an + 1
a1 ~ a7 = ?
This sequence is the Fibonacci Sequence.

Solution

Mathematical Induction

Definition

1. Show that n = 1 is true.
2. Assume n = k is true.
3. Show that n = k + 1 is true.
A mathematical induction is a way
to prove the given statement (given).
1. Show that (given) is true
when n = 1.
2. Assume that (given) is true
when n = k.
3. Use the (given) when n = k
to show that (given) is true
when n = k + 1.
Then, just like a recursive formula,
(given) is true.
(given) is true when n = 1.
→ (given) is true when n = 1 + 1 = 2.
→ (given) is true when n = 2 + 1 = 3.
...

Example

Prove the given statement.
(n is a natural number.)
1 + 2 + 3 + ... + n = n(n + 1)2
Solution