Sum Of Ten Successive Fibonacci Numbers

                                     

Sum of any ten successive Fibonacci numbers is eleven times the seventh number.

Observe the following sequence.
1,1,2,3,5,8,13,21,34,55,89,…

In this sequence first two numbers are 1and 1.Third number is the sum of the first two numbers. Fourth number is the sum of the second and the third number. Each subsequent number is the sum of the two numbers before it. This sequence is called Fibonacci number sequence.
 

Fibonacci was an Italian mathematician. He introduced the Hindu-Arabic numeration system
to the west countries. He is well known in the modern world for spreading the Hindu-Arabic
Numeration system. The sequence of numbers is named after him. In Fibonacci sequence the first two numbers are 1 and 1.It is not necessary to take first two numbers be 1and 1.Take any two numbers then form Fibonacci like sequence.

For example
5, 7, 12, 19, 31, 50, 81…

Consider the Fibonacci number sequence of 10 numbers.
1+1+2+3+5+8+13+21+34+55

In this sequence seventh number is 13 and the sum of all ten numbers is 143.143 is the product of 13 and 11.

This rule is applicable for any ten Fibonacci like sequence.

Another example:
5,7,12,19,31,50,81,131,212,343.
In this sequence seventh number is 81.
81x11=891.
Sum of all ten numbers is also 891.

How does this work?
Suppose a and b are any two numbers.
The sequence will be as follows,

a
b
a+b
a+2b
2a+3b
3a+5b
5a+8b
8a+13b
13a+21b
21a+34b

------------
55a+88b

Here seventh number is 5a+8b, when it is multiplied by 11 we get 55a+88b which is the total of all ten numbers.