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 × 7th number.
Taking a = 8, b = 13; write 10 Fibonacci numbers and verify that sum of all these numbers = 11 × 7th number.
Solution:
Given:
$a=8$ and $b=13$
The numbers in the Fibonnaci sequence are arranged in the following manner:
$1 s t, 2 n d,(1 s t+2 n d),(2 n d+3 t h),(3 t h+4 t h),(4 t h+5 t h),(5 t h+6 t h),(6 t h+7 t h),(7 t h+8 t h),(8 t h+9 t h),(9 t h+10 t h)$
The numbers are $8,13,21,34,55,89,144,233,377$ and 610 .
Sum of the numbers $=8+13+21+34+55+89+144+233+377+610$
= 1584
$11 \times 7$ th number $=11 \times 144=1584$