Question

Fibonacci numbers Take 10 numbers as shown below:

a, b, (a + b), (a + 2b), (2a +3b), (3a + 5b), (5a +8b), (8a +13b), (13a + 21b), and (21a +34b).

Sum of all these numbers =11(5a + 8b) =11 $\times$ 7th number.

Taking a = 8, b= 13; write 10 Fibonacci numbers and verify that sum of all these numbers =11 $\times$ 7th number.

Answer 1

The given numbers are

a, b (a + b), (a + 2b), (2a + 3b), (3a + 5b), (5a + 8b), (8a + 13b), (13a + 21b), and (21a + 34b).

Sum of there numbers = 11 (5a + 8b) = 11 $\times$ 7th number

Now taking a = 8, b = 13, then the 10 number be 8, 13, 21, 34, 55, 89, 144, 233, 377, 610

Whose 7th number = 144

By adding these 10 numbers, we get the sum

= 8 + 13 + 21 +34 + 55 + 89 + 144 + 233 + 377 + 610 = 1584

and 11 $\times$ 7th number = 11 $\times$ 144

= 1584

Which is same in each case