The table below shows the terms F0−F15 in the Fibonacci sequence. nFn001121324355687138219341055118912144132331437715610 The Fibonacci sequence is defined formally by setting F0=0,F1=1, and then later terms are defined by the recursive formula Fn−1+Fn=Fn+1,∀n≥1.(∗∗) Note that what this formula states that if two consecutive terms in the Fibonacci sequence are added, then the sum equals the next term. For example, if n=12, it states that F12−1+F12=F12+1, or F11+F12=F13, and this can be seen to be true from the table: 89+144=233.

Show that the identity F2+F4+F6+⋯+F2n=F2n+1−1(∗) holds for n=1,2,3,4 by computing each side of (*) separately for those values of n and show that they are equal.

When n=1 the left side of () =F_ and the right side of () = F_ −1. After putting in the values from the tables, this states that the left side of () = and the right side of () = −1= , and so both sides are equal.

When n=2 the left side of () =F_ +F_ and the right side of () = F_ −1. After putting in the values from the tables, this states that the left side of () = + = , and the right side of () = −1= , and so both sides are equal.

When n=3 the left side of () =F_ +F_ +F_ and the right side of () = F_ −1. After putting in the values from the tables, this states that the left side of () = + + = , and the right side of () = −1= , and so both sides are equal.

When n=4 the left side of () =F_ +F_ +F_ +F_ and the right side of () = F_ −1. After putting in the values from the tables, this states that the left side of () = + + + = , and the right side of () = −1= , and so both sides are equal.