Math Problem Statement

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.

Solution

We need to verify the identity

F_2 + F_4 + F_6 + \dots + F_{2n} = F_{2n+1} - 1 \tag{*}

for n=1,2,3,4n = 1, 2, 3, 4, by calculating each side separately and checking if they are equal.

Step 1: Verifying for n=1n = 1

Left Side:

For n=1n = 1, we only have the term F2F_2. Left side=F2=1\text{Left side} = F_2 = 1

Right Side:

For n=1n = 1, we have Right side=F2(1)+1−1=F3−1=2−1=1\text{Right side} = F_{2(1) + 1} - 1 = F_3 - 1 = 2 - 1 = 1 Both sides are equal.

Step 2: Verifying for n=2n = 2

Left Side:

For n=2n = 2, we need to sum F2+F4F_2 + F_4: Left side=F2+F4=1+3=4\text{Left side} = F_2 + F_4 = 1 + 3 = 4

Right Side:

For n=2n = 2, we have Right side=F2(2)+1−1=F5−1=5−1=4\text{Right side} = F_{2(2) + 1} - 1 = F_5 - 1 = 5 - 1 = 4 Both sides are equal.

Step 3: Verifying for n=3n = 3

Left Side:

For n=3n = 3, we sum F2+F4+F6F_2 + F_4 + F_6: Left side=F2+F4+F6=1+3+8=12\text{Left side} = F_2 + F_4 + F_6 = 1 + 3 + 8 = 12

Right Side:

For n=3n = 3, we have Right side=F2(3)+1−1=F7−1=13−1=12\text{Right side} = F_{2(3) + 1} - 1 = F_7 - 1 = 13 - 1 = 12 Both sides are equal.

Step 4: Verifying for n=4n = 4

Left Side:

For n=4n = 4, we sum F2+F4+F6+F8F_2 + F_4 + F_6 + F_8: Left side=F2+F4+F6+F8=1+3+8+21=33\text{Left side} = F_2 + F_4 + F_6 + F_8 = 1 + 3 + 8 + 21 = 33

Right Side:

For n=4n = 4, we have Right side=F2(4)+1−1=F9−1=34−1=33\text{Right side} = F_{2(4) + 1} - 1 = F_9 - 1 = 34 - 1 = 33 Both sides are equal.

Conclusion:

For n=1,2,3,4n = 1, 2, 3, 4, we have verified that the identity F2+F4+F6+⋯+F2n=F2n+1−1F_2 + F_4 + F_6 + \dots + F_{2n} = F_{2n+1} - 1 holds true by computing both sides and confirming that they are equal.

Would you like further details on this or have any other questions?

Here are 5 questions to expand this concept:

  1. How can this identity be proved for all values of nn using mathematical induction?
  2. What are the implications of this identity in terms of Fibonacci sequence properties?
  3. Can this sum of even-indexed Fibonacci numbers be related to other known Fibonacci identities?
  4. How does this identity behave for larger values of nn?
  5. Can this identity be visualized geometrically, perhaps as sums of areas or distances?

Tip: The Fibonacci sequence has many fascinating properties, including relationships to the golden ratio and sums of subsets of Fibonacci numbers. Exploring these can lead to deeper insights into number theory!

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Math Problem Analysis

Mathematical Concepts

Number Theory
Fibonacci Sequence
Summation Identities

Formulas

Fibonacci Sequence: F_n = F_{n-1} + F_{n-2}
Summation Identity: F_2 + F_4 + ⋯ + F_{2n} = F_{2n+1} − 1

Theorems

Fibonacci Summation Identity

Suitable Grade Level

Grades 9-12

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