Question

The following summation are Fibonacci numbers starting with F₁ = 1. ( that is 1, 1, 2, 3, 5........):-

F₁ + F₂ + F₃ + .......... + Fₙ = Fₙ + 2.
Is this True or False?
If false how can you change it so it's true?

Answer 1

`\textsf{the sum is }F_(n+2)-1`

Explanation:

The given statement is FALSE.

Consider ...

`F_(n+2)=F_(n+1)+F_n\\\\F_n=F_(n+2)-F_(n+1)\\\\\displaystyle \sum_(i=1)^n{F_i}=\sum_(i=1)^n{F_(i+2)}-\sum_(i=1)^n{F_(i+1)}=F_(n+2)+\sum_(i=3)^(n+1)(F_i-F_i)-F_2\\\\\sum_(i=1)^n{F_i}=F_(n+2)-F_2 =\boxed{F_(n+2)-1}`