Algebra II/Sequences and Series

What is a sequence?

A sequence can be best described as a sequence of numbers which follow a pattern. For example, 1,2,3,4 is a sequence where we keep adding 1 to the term. 1,1,2,3,5,.. is also a sequence, a special type called the Fibonacci Sequence where we add the previous number. A sequence must increase or decrease in size. For example, 1,1,1,1,1... is not a sequence since there is no change (adding or subtracting 0 is not valid).

We call a number that is part of a sequence as a term. Each term also has a number assigned to it, example 1st term, 2nd term etc. Lets take the sequence:

$1,3,5,7,9,...$

There are 5 terms shown. Now, we can find what term number each number is. For example, $3$  is the 2nd term of the sequence which we express as $t_{2}$ . $9$  is the 5th term of the sequence which we express as $t_{5}$ . The idea of terms is very important to understanding sequences and series.

Recursive formula

Most sequences can be expressed in terms of a recursive formula that is unique to each sequence. We write the recursive formula in the general form:

$t_{1}=a,t_{n+1}=rt_{n}+d$

where $a$  is the first term ($t_{1}$ ) and $r$  and $d$  are the common constant that we add and/or multiply to our current term ($t_{n}$ ) to get our next term, ($t_{n+1}$ ). For example, take the sequence:

$4,9,14,19,24,29,...$

$t_{1}$  of this sequence is $4$ . Great, we've got our first part. Now to figure out the common difference. It seems that we're adding a constant number each time to our term. Take the difference between any 2 consecutive (one after the other) terms,

Sequences have a common pattern and the way they increase or decrease is based upon the type of sequence. There are two types of sequences: arithmetic and geometric, which increase in different ways to each other.

Arithmetic

The arithmetic sequence is a sequence of numbers where the previous number is added to by a common difference, to get the next term. For example, 1,2,3,4... and 2,5,8,11... are both examples of arithmetic sequences, where we are adding (subtracting can be expressed as adding a negative number) a constant term to get our next term. The general recursive formula for an arithmetic sequence is:

$t_{1}=a,t_{n+1}=t_{n}+d$

Where, $a$  is the starting term and $d$  is the common difference between two consecutive terms. We can figure out the common difference of an arithmetic sequence simply by doingː

$d=t_{n+1}-t_{n}$

An arithmetic sequence can also be expressed using an explicit formula, that is a formula which we can use to find the value of any term in our sequence. This is given by:

$t_{n}=a+d(n-1)$

Where again, $a$  is the starting term, $d$  is the common difference and $n$  is the term number. Using the explicit formula allows us to quickly figure out any term number without writing out the whole sequence. Finally, we can figure out an arithmetic mean, which is pretty much the number between two terms. Suppose we have terms $a,b,c$  all in a consecutive sequence. If we know $a$  and $c$ , we can figure what $b$  is simply by doingː

$b={\frac {a+c}{2}}$

Examples

Lets take the sequence $1,4,7,10,13,16...$ . We first need to figure out if this is an arithmetic sequence. Take 2 pairs of numbers, $4,7$  and $10,13$ . $7-4=3$  and $13-10=3$ , so the common difference $d$  is 3 and thus it is an arithmetic sequence. Now we can easily buildup a recursive formula and our explicit formula. Remember that the recursive formula for a sequence is given byː

$t_{1}=a,t_{n+1}=t_{n}+d$

Since we already know that $d=3$ , working out $a$  is pretty simple, since that's our first term which is $1$ . We can also use this information in our explicit formulaː

$t_{n}=a+d(n-1)$

Thus we get the two different formulas for our sequenceː

$t_{1}=1,t_{n+1}=t_{n}+3$

$t_{n}=1+3(n-1)$

And now we can do many things with this.

What is the 20th term of the sequence?

Well, by using our explicit formula and setting, $n=20$ , we getː

$t_{n}=1+3(20-1)$

$=1+3\times 19$

$t_{20}=58$

If $t_{20}=58$  and $t_{2}=4$ , what is the value for $t_{11}$ ?

Now we can brute force this by using our explicit formula, BUT there is a much cleverer method. And that method is using our arithmetic mean. If we add 20 and 2, we get 22 and dividing it by 2 we get 11, term 11 is the mean of $t_{20}$  and $t_{2}$ . So we use our arithmetic mean formulaː

$b={\frac {a+c}{2}}$

Subbing in our valuesː

$b={\frac {58+4}{2}}={\frac {62}{2}}=31$

So $t_{11}=30$ . We can use our explicit formula to verify our answer.

$t_{n}=1+3(11-1)$

= $1+3\times 10=31$

And thus we got our answer the clever way.

Geometric

A geometric sequence is a sequence which instead of adding, we are multiplying by a constant value, which we call the common ratio, denoted by $r$  (division can be interpreted as multiplying by fractions). e.g. 2,4,8,16,... is an example of a geometric sequence (we are multiplying by 2 every time) so is 4,2,1,0.5,0.25... (we are multiplying by 0.5 every time). Now, the recursive formula of a geometric sequence