Expressing Fibonacci Sums in Terms of Fibonacci Numbers

Let’s approach this problem step by step, expressing each result in terms of Fibonacci terms.

a)

To solve this, we can use the formula for the sum of Fibonacci numbers up to the nth term:

For , we get:

b)

Using the same formula as above, with :

c)

This is the sum of odd-indexed Fibonacci numbers up to . There’s a formula for this:

Here, , so . Therefore:

d)

This is the sum of even-indexed Fibonacci numbers up to . The formula for this is:

Here, , so . Therefore:

e)

This is the sum of Fibonacci numbers with indices that are multiples of 3 plus 1. There’s no simple closed form for this sum in terms of a single Fibonacci number.

f)

Similarly, this is the sum of Fibonacci numbers with indices that are multiples of 4 plus 1. There’s no simple closed form for this sum in terms of a single Fibonacci number.

Final answers:
a)
b)
c)
d)
e) No simple closed form
f) No simple closed form