Sequences and Series TUTORIAL!!!!!!!!!

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anonymous · Answer

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Sequences are very important in solving many problems. Sequences can be arithmetic, geometric, both and none of these. Sequences also can have the first term or not have it and can be finite or infinite. If 
the sequence is finite then it has a last term, and if it doesn't then it is infinite. We will discuss sequences that have first term. Arithmetic sequence is the sequence that can be written by adding or subtracting the certain number from the previous term. Mathematically they are written as 
$\large\color{black}{a_n=a_1+d(n-1)}$, where $\large\color{black}{a_n}$ the nth term of sequence, $\large\color{black}{a_1}$ is first term and d is the difference between terms. Arithmetic sequences can be determined by finding the difference between any two consecutive terms and checking if this difference works for any two other consecutive terms of sequence. For example, the sequence 1, 2, 3, 4 ... is arithmetic because the difference between any two consecutive terms is 1.
Lets write a formula for sequence 2, 5, 8, 11, 14 ... .
First we find the first term. In the given sequence it is 2. So a_1=2.
Then we find the difference between consecutive terms. In this sequence the difference is 3.
Finally, the formula is $\large\color{black}{a_n=2+3(n-1)}$.
Examples of arithmetic sequences:
a) 1, -3, -7, -11, ...
b) 0, 10, 20, 30, ...
c) 2, 2, 2, 2, ...

Now lets see what is geometric sequence is. Let our sequence be 2, 4, 8, 16, 32 ... . The difference between consecutive terms is not the same, so it is not arithmetic sequence. But we can see that if we 
divide consecutive terms, like 4/2, 8/4, 16/8, 32/16 ... , we will get everywhere 2. This type of sequences is called geometric sequence. Mathematically it is written as 
$$b_n=b_1*q ^{n-1}$$
where $\large\color{black}{b_n}$ is nth term of sequence, $\large\color{black}{b_1}$ is first term of sequence and q is quotient between two consecutive terms. Geometric sequences can be determined by finding the quotient of each pair of consecutive terms and checking if it is the same. For example, the sequence 1, 5, 25, 125, ... is geometric because the quotient is 5.
Lets write the formula for sequence 6, 12, 24, 42, ... .
First we will find the first term. In this sequence it is 6.
Then we will find the quotient. In this sequence it is 2.
Finally the formula will be $$b_n=6*2^{n-1}$$.
Examples of geometric sequences:
a) 1, 7, 49, 343, ...
b) 20, 10, 5, 2.5, ...
c) 10, 10, 10, 10 ...

We have discussed geometric and arithmetic sequences, but can there be a sequence that is both geometric and arithmetic? Yes, if we have a sequence k, k, k, k, ..., for which the d is 0 and q is 1.
But there are also sequences that are neither arithmetic neither geometric. For example the well-known Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, ... . It doesn't have nor same d nor same q.

The sequence shown above is Fibonacci sequence and also have the way to be defined. It is not defined in formulas we saw at arithmetic and geometric sequences. This way to define the sequence is called recursively defined sequence. This means that the term in sequence is defined based on previous term or terms. Geometric and arithmetic sequence can be defined recursively, but we will do it later.
Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, ... .
Let's see if there is some relationship between terms.
In this sequence the next term equals to sum of two previous. So we can write it as 
$$a_{n+1}=a_n + a_{n-1}$$. In recursive definition of sequence we must include the value of first term(s).
$\large\color{black}{a_{n+1}=a_n + a_{n-1}}$, where $\large\color{black}{a_1=a_2=1}$.
For example, define the sequence -4, 4, -4, 4, -4, 4 recursively.
First we will look at our sequence and look for any relationships between terms of sequence. We can see that the first term is -4 and second is 4 and then again -4 and then 4. So the term of sequence, where n>2, will be defined as $\large\color{black}{a_n=a_{n-2}}$.
Finally the formula is $\large\color{black}{a_n=a_{n-2}}$, where $\large\color{black}{a_1=-4}$ and $\large\color{black}{a_2=4}$.
Examples of recursively defined sequences:
a) -1, 1, -1, 1, -1, 1
b) 0, 5, 15, 35, 75
To write a formula for (b) w can do the following:
1) divide 15/5, 35/15 and 75/35. We can see the whole part is always 2. So we can think of 
$\large\color{black}{a_n=2*a_{n-1}+b}$. To find b we can choose for example 5 and 15 and find b at which 15=2*5+b.
After we find the b, we need to check the formula. The final formula is $\large\color{black}{a_n=2*a_{n-1}+5}$.
Now lets define arithmetic and geometric sequence recursively:
Arithmetic sequence: $\large\color{black}{a_n=a_{n-1}+d}$
Geometric sequence: $\large\color{black}{b_n=b_{n-1}*q}$

There are sequences that are not defined recursively but are not geometric or arithmetic. For example, $\large\color{black}{a_n=n^2}$, or $\large\color{black}{a_n=1/n}$.

Now lets discuss one big thing about geometric and arithmetic sequences. How do we find the sum of terms in the arithmetic sequence? How do we find the sum of terms in the geometric sequence?
Arithmetic sequence:
$\large\color{black}{S_n=(a_1+a_n)*n/2}$
Geometric sequence:
$\large\color{black}{S_n=b_1*(q^n-1)/(q-1)}$

There is one big thing about geometric sequences. For example we have a sequence 4, 2, 1, 0.5, 0.25, 0.125, ... . We can find its sum of terms, no matter that it is infinite.
$\large\color{black}{S_n=b_1/(q-1)}$

Lets practice:
Determine if the sequence is geometric, arithmetic, both or neither:
a) -1, 1, -1, 1, -1, 1, ...
b) 2, 4, 6, 8, 10, 12, ...
c) 0.1, 0.01, 0.001, 0.0001, ...
d) 0, 5, 6, 11, 12, 17, 18, ...
e) 0.23, 0.24, 0.25, 0.26, 0.27, ...
f) 0.3, 0.3, 0.03, 0.03, 0.003, 0.003, ...
g) 10, -20, 40, -80, 160, -320, ...
h) 0, 0, 2, 2, 4, 4, 6, 6, 8, 8, ...
i) 87, 89, 91, 93, 95, 97, ...
j) 100001, 10000.1, 1000.01, 100.001, 10.0001, 1.00001, 0.100001, ...

Answers:
a) This is geometric sequence. The quotient is -1.
b) This is arithmetic sequence. The difference is 2.
c) This is geometric sequence. The quotient is 0.1.
d) This is neither geometric neither arithmetic sequence.
e) This is arithmetic sequence. The difference is 0.01.
f) This is neither geometric neither arithmetic sequence.
g) This is geometric sequence. The quotient is -2.
h) This is neither arithmetic neither geometric sequence.
i) This is arithmetic sequence. The difference is 2.
j) This is geometric sequence. The quotient is 0.1.

thesmartone · Answer

Good job :D

preetha · Answer

@abhisar

abhisar · Answer

Neat and very informative !
Thanx for sharing with us :-)

solomonzelman · Answer

Maybe perhaps add about  convergence/diverges  infinite series and sequences.

anonymous · Answer

thnx for advice. i will do it next time.

dan815 · Answer

It would be great if there a tutorial section where all this stuff can be added, I can just reference people to this stuff