Component of Fibonacci Sequence

[leospyder]

Please excuse my total ignorance but can someon explain to me how the following part of a certain proof makes sense

We want to show that Fk+1 ≤ (7/4)^(k+1). Consider fk+1 = fk + fk−1 (We can do this
as k +1 is at least 2; see the comment following the basis) < (7/4)^k +(7 /4)^(k−1) (by the Induction Hypothesis;
notice how the stronger hypothesis comes in handy here.)

The parts I bolded in red are mainly the things I don't understand. I plugged in the (7/4)...part into my calculator and did not get the alleged answer I was supposed to get if it were simply (7/4)^k+1. Can someone please enlighten me?

 

[matt grime]

If a<b, and c>0, then ac<bc (and no that is not cryptic - you end up with something that you wish to show is less than something else - there is no reason to suppose they are equal, nor is it necessary. If I want to show something is less than 4 and I show it is less than 3 I've shown it is less than 4, for example).

 

[tacman]

The Fibonacci sequence is a famous sequence of numbers where each number is the sum of the two previous numbers, starting with 0 and 1. So the sequence goes like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

The proof is trying to show that a certain property holds for all numbers in the Fibonacci sequence. In this case, the property is that Fk+1 (the k+1th number in the sequence) is less than or equal to (7/4)^(k+1). In other words, the numbers in the Fibonacci sequence are growing at a rate that is less than or equal to (7/4)^(k+1).

To prove this, the proof starts by considering the formula for the k+1th number in the Fibonacci sequence: fk+1 = fk + fk−1. This is because the k+1th number is just the sum of the previous two numbers.

Next, the proof uses the Induction Hypothesis, which is a technique commonly used in mathematical proofs. The Induction Hypothesis states that if a property holds for a certain number (in this case, k), then it also holds for the next number (k+1). So, since the property holds for k, it also holds for fk and fk-1. Therefore, we can write fk+1 as (7/4)^k +(7 /4)^(k−1), using the Induction Hypothesis.

Finally, the proof uses the fact that the Induction Hypothesis is a stronger hypothesis. In this case, it means that the property holds not only for k, but also for all numbers before k. This allows us to rewrite fk+1 as (7/4)^k +(7 /4)^(k−1), which is a stronger statement than just (7/4)^k+1.

You may have plugged in (7/4)^k+1 into your calculator and not gotten the alleged answer because the proof is not claiming that fk+1 is equal to (7/4)^k+1. It is claiming that fk+1 is less than or equal to (7/4)^k+1. In other words, the proof is showing that the growth rate of the Fibonacci sequence is less than or equal to the growth rate of (7