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6. Geometric sequences

Geometric sequences

One should be aware of the meaning of the term "arithmetic sequence". The term "geometric term" is a similar one, and refers merely to a constant multiplication (as opposed to constant addition or subtraction).

In general, when one multiplies by a constant in order to obtain the next value of a sequence, the sequence is called a geometric sequence.

For example:

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, ...

That was a geometric sequence. One can tell that there is a multiplication of 2 to get each term, and hence one can use a recursive definition:

Where .

One other point to consider is that the sequence has very large numbers in after a short time, even though the multiplication is only by 2 each time.

In general, a geometric sequence is defined (recursively) as:

.

The reason for using "i" as opposed to "r" for the terms in the sequence is simply that "r" is used to refer to the common ratio, this is the number which one multiplies any given term in the sequence by in order to obtain the next.

It is rather simple to formulate an expression for the "ith term" of a geometric sequence. Consider that one is multiplying each term by "r", beginning at ; hence:

.

(Where "a" deontes the first term of the sequence).

Example

1. Which term of the following sequence is 65536:

Hence:

Therefore it is the 9th term.

Summing geometric series

As with arithmetic sequences, one is often required to sum all of the terms (up to a limit) of a geometric sequence. One refers (as with the arithmetic equivalent) to these summations as summations of geometric series.

Consider:

Now, if one is to multiply by "r":

These equations have many terms in common, such that:

Therefore:

Hence:

Now one has a method to sum the first "n" terms of a geometric sequence.

Example

1. Sum the first 10 terms of a sequence in which:

,

and:

.

This is a simple case of using the fomula.

Convergent geometric series

One might have noticed that if one has a common ratio that is less than 1, but greater than 0, the terms will get successively smaller, but never negative, this is the concept of convergence.

First one must define a term: sum sequence.

For any sequence, one can make another sequence by summing the terms of the original sequence.

Let "s" denote the sum sequence of the sequence "u":

Consider also, that the sum sequence can be defined recursively:

.

The sum sequence of a sequence which has a common ration which is , will tend towards a value. If the common ratio is greater than 1, the sequence will continue to get larger but the sum sequence will not tend towards anything (other than infinity).

A sequence which does not have a sum sequence that tends towards a finite value is said to be divergent (the sequence diverges to infinity).

If the sum sequence does tend towards a finite value, the sequence is said to be convergent, and one can give the value to which the sequence will converge.

Some sequences have a common ratio that is negative. This will cause them (if the negative value is such that ) to converge. If the common ratio is less than -1, the sequence will oscillate infinitely (as the value of the sequence will become positive and negative alternately, while becoming infinitely large).

Consider now that one can calculate the value to which the sum sequence converges.

If , and (), , hence the sum sequence will converge at:

.

One refers to this convergence of the sum sequence as the sum to infinity:

.

Using sigma notation

One can represent the sum to infinity using sigma notation:

One can use an upper limit of a variable, and then limit the variable to infinity, however the above notation is also valid.

Also See

Read these other OCR Core 2 notes:

1. Trigonometry
2. Sequences
3. The binomial theorem
4. The sine and cosine rules
5. Integration
6. Geometric sequences
7. Exponentials and logarithms
8. Factors and remainders
9. Radians
10. The trapezium rule