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Arithmetic and Geometric Sequences

Adding d again to a(3) we get a(4).

a(4) = a + 3*d

a(5) = a + 4*d

a(6) = a + 5*d

a(7) = a + 6*d

etc.

We can now see a pattern. The nth term in the sequence is always a(1) (=a) plus n-1 times our common difference, d. Using this, we can now write

a(n) = a + (n-1)*d

This gives us our general form of the arithmetic sequence! Now we can start doing some more complicated examples.

Example 2: We are given the following sequence.

3, 8, 13, 18, 23, .......

Find a(1). What is the common difference? What is the general formula for this sequence? Find a(34).

First, solve this out yourself. Then look at the solution.

Solution:

a(1) = 3

This comes straight from 3 being the literal first term in the sequence.

The common difference is the difference (or distance, whichever floats your boat) between two consecutive terms in the sequence.

d = 18 - 13 

d = 5

We use our general form the arithmetic sequence as we derived above.

a(n) = a + (n-1)*d

Now, we just plug in our a (=a(1)) and d values.

a(n) = 3 + (n-1)*5

We can simplify this to get a nicer looking answer.

a(n) = 3 + 5*n -5

a(n) = 5*n - 2

This is our general formula for this sequence.

The 34th term of this sequence is

a(34) = 5*34 - 2

a(34) = 170 - 2

a(34) = 168

This finishes everything we can do with arithmetic sequences. Now let's move onto geometric sequences.

For geometric sequences, the next term of the sequence is always the previous term multiplied (or divided) by the same value. The value multiplied or divided is known as the common ratio, typically denoted by "r." We can get this value by dividing consecutive terms in the sequence. The geometric sequences will be denoted by g(n) or "g subscript n." Our first term in the sequence is denoted by g(1) (=g). With this, we can now derive the general formula of a geometric sequence.

Let's start by listing out the terms we know.

g(1) = g

Now, let's create g(2) by multiplying g(1) by some value, r (could be an integer or a fraction).

g(2) = g*r

Let's repeat the same process by multiplying g(2) by r to create g(3), and so on and so forth.

g(3) = g*r*r

g(3) = g*(r^2)

g(4) = g*(r^2)*r

g(4) = g*(r^3)

g(5) = g*(r^4)

g(6) = g*(r^5)

etc.

From here, we can see the pattern that each term of the sequence is g (=g(1)) times some power of r. Not only that, but we can also see that the power r is raised to is the position of the number in the sequence minus 1. This allows us to write down the general formula for the nth term of a geometric sequence.