A sequence, in mathematics, is a string of objects, like numbers, that follow a pattern. The individual elements in a sequence are called terms.

Number sequence questions as for a particular term in the sequence, which can be calculated or reasons from the pattern.

Number sequences are College Level Math and appear on standardized tests like the Accuplacer and the COMPASS.

A sequence of numbers is a set of numbers, but here they are in order. For example, we can represent the set of natural numbers N as a sequence 1, 2, 3,… A sequence can be finite or infinite. In our case of the sequence of the natural numbers, we have an infinite sequence.

If we have a sequence of numbers a₁, a₂, a₃, … we denote that sequence by {a_n}. We can write, for example, the sequence of natural numbers like this:

a_n = a_n-1 + 1 or a_n+1 = a_n+1

From this formula, we can see that each number is greater than the previous number by one, which is true for the sequence of the natural numbers.

The first term (member) of the sequence is denoted by a₀. So, if we know the first term of the sequence and we know the formula that describes the sequence, we can find any term of that sequence. Even if we know some other member of the sequence, we can find other members.

Let’s solve 2 examples for both cases:

1) If a₀ = 2 and an = an-1 – 2, find the 4th member of the sequence {a_n}.
Let’s find 2nd and 3rd member, which we will use to find the 4th.

a₁ = a₀ – 2 = 2 – 2 = 0
a₂ = a₁ – 2 = 0 – 2 = -2
a₃ = a₂ – 2 = -2 – 2 = -4

So, our 4th member is number -4.

2) If a₂ = 4 and a_n = 2a_n-1, find the 1st member of the sequence {a_n}.

a₂ = 2a₁ → 4 = 2a₁ → a₁ = 2
a₁ = 2 a₀ → 2 = 2a₀ → a₀ = 1

So, our first member is 1.

Practice Questions

1. If a₀ = 3 and a_n =  – a_n-1 + 3, find a₃ of the sequence {a_n}.

A. 0
B. 1
C. 2
D. 3

2. If terms of the sequence {a_n} are represented by a_n =  a_n-1/n and a₁ = 1, find a₄.

A. 1/2
B. 1/4
C. 1/16
D. 1/24

3. If a₀ = 1/2 and a_n = 2a_n-1² , find a₂ of the sequence {a_n}.

A. 1/2
B. 1/4
C. 1/16
D. 1/24

4. If members of the sequence {a_n} are represented by a_n =  (-1)ⁿa_n-1 and if a₂ = 2, find a₀.

A. 2
B. 1
C. 0
D. -2

5. If the sequence {a_n} is defined by a_n+1 =  1- a_n and a₂ = 6, find a₄.

A. 2
B. 1
C. 6
D. -1

6. If members of the sequence {a_n} are represented by a_n+1 =  – a_n-1 and a₂ = 3 and, find a₃ +  a₄.

A. 2
B. 3
C. 0
D. -2

1. A. 0

a₀ = 3

a_n =  – a_n-1 + 3

a₁ =  – a₀ + 3 = -3 + 3 = 0

a₂ =  – a₁ + 3 = 0 + 3 = 3

a₃ =  – a₂ + 3 = -3 + 3 = 0

2. D. 1/24

a_n =  a_n-1/n

a₁ = 1

a₂ =  a₁/2 = 1/2

a₃ =  a₂/3 = (1/2)/3 = 1/6

a₄ =  a₃/4 = (1/6)/4 = 1/24

3. A. 1/2

a₀ = 1/2

a_n = 2a_n-1²

a₁ = 2a₀² = 2·(1/2)² = 2·(1/4) = 1/2

a₂ = 2a₁² = 2·(1/2)² = 2·(1/4) = 1/2

4. D -2

a_n= (-1)ⁿa_n-1

a₂=2

2=a₂= (-1)²a₁= a₁  ?  a₁=2

a₁= (-1)¹a₀

2=- a₀

a₀=-2

5. C. 6

a_n+1 =  1 –  a_n

a₂ = 6

a₃ = 1 – a₂ = 1 – 6 = -5

a₄ =  1 –  a₃ = 1 – (-5) = 1 + 5 = 6

6. C. 0

a_n+1 = – a_n-1

a₂ = 3

a₃ =  – a₂ = -3

a₄ =  – a₃ = -(-3) = 3

a₃ + a₄ =  -3 + 3 = 0