Arithmetic and Geometric Sequences

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each successive number is obtained by adding a fixed number, called the common difference, to the previous number. For example, the sequence 5, 10, 15, 20, … is an arithmetic sequence with common difference 5. The general form of an arithmetic sequence is

a, a+d, a+2d, … ,
where a is the first term and d is the common difference.

Geometric Sequences

In a geometric sequence each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54 … is a geometric sequence with common ratio 3.

The nth term of a geometric sequence with first term a and common ratio r is given by: a n = ar(n−1)

so in our example above we have: 2 3(n−1) = 6n−3
Substituting n = 13 gives us: 2 3(13−1) = 6(13)−3 = 6 × 10 = 60
Therefore, the thirteenth term of our sequence is 60.

Finding the nth Term

There are a few ways to find the nth term of a sequence. In this example, we can see that the sequence is doubling each time. So, we can just double the previous term to find the next term. 8 * 2 = 16, 16 * 2 = 32, 32 * 2 = 64. So, the thirteenth term would be 64 * 2 = 128.

Arithmetic Sequences

In an arithmetic sequence, each term after the first is obtained by adding a fixed constant, d, to the preceding term.

a_n=a_1+d(n-1)

where a_n represents the nth term of the sequence, a_1 represents the first term in the sequence, and n is the number of terms in the sequence.

The common difference (d) can be represented as: d=a_n-a_1

Geometric Sequences

In a geometric sequence each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, the sequence 3, 6, 12, 24,… is a geometric sequence with common ratio 2.

The nth term of a geometric sequence with first term a1 and common ratio r is given by:
an=a1rn−1

In the example above:
a1=3
r=2
an=32(13−1)=32×2^12=40960

Application

Expressions can be written in various forms to give the thirteenth term of a sequence. For example, the thirteenth term of the sequence 8, 16, 32, 64 can be given as 13=8*2^(13-1).

Arithmetic Sequences

In an arithmetic sequence, each term after the first is obtained by adding a constant, d, to the preceding term. The common difference, d, is equal to the difference between any two consecutive terms in the sequence. Therefore, we can find d by subtracting one term in the sequence from another term. In this sequence, d=16-8=8. We can use this to find the thirteenth term:

a13=a1+12d
a13=8+12(8)
a13=8+96
a13=104

Geometric Sequences

In mathematics, a geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54,… is a geometric sequence with common ratio 3.

Geometric sequences are characterized by either having all terms as non-negative real numbers, or all terms as positive real numbers. They are sometimes also called finite geometric progressions (FGPs), since they may be described as progressions where each successive entry is obtained by multiplying the previous entry by some constant value (called the common ratio), and then dropping the zero terms (which do not exist in an FGP).