Basic Fibonacci Sequences

Problem Level: 1

Find the 8^th term of the Fibonacci sequence.

Solution:

For those that do not know this sequence; here it is;

0,1,1,2,3,5,8,13,21 ...

Starting with zero and to get the next term you add the previous 2 terms.

The eighth term of the sequence would be:

0, 1 ,1 ,2 ,3 ,5, 8, 13?

The first term of this sequence is one so the correct answer would be: 21.

The "zero" term is zero.

Combined Sequences

Problem Level: 4 

Find the sum of this unique sequence: 1,2,5,8,8,11,14,15,17, 20, 22,23... 121

Hint:

I have decided not to put a solution so I can challenge some people.  Only use the hint when you cannot find a solution.

We see no real pattern in the sequence from one number to another.  Many questions form when two eights show.  This tells us that there is a +0 somewhere. 

After much analysis, most people will give up.

What you need to know is to look at patterns between numbers, not necessarily at consecutive numbers.

But what these people overlook is that this sequence is made up of two sequences:

1,8,15,22... (up to 121) and

2,5,8,11,14,17,20,23... 121

After we have successfully decoded the message, the rest is up to you to finish it.

Arithmetic Sequence

Problem Level: 3

Find the sum of the arithemetic sequence -2, 4, 10 ... 100.

Solution:

The sum of any arithmetic sequence is the number of numbers in the sequence times the average of the sequence. 

The average (arithmetic mean) of the sequence is very easy to find.

(-2 +100 ) /2

=98/2

=49

The average of the sequence is 49.

To find the number of terms can be a bit more challenging.

Because the common difference is 6, try to make the first term 6.

We can do this by adding eight.  The new sequence is  6, 12 ,18 ...108

Dividing each term by 6 leaves us with: 1,2,3... 53.

This shows that there are 53 numbers in this sequence

53 x 49 = 2597

Answer: The sum of the sequence is 2597

We used counting techniques and arithmetic mean to solve this problem.  Most problems require many types of math to solve.