If fib (22) = 17,711 and fib (24) = 46,368, what is fib (23)?
The Correct Answer and Explanation is:
To find fib(23), given:
- fib(22) = 17,711
- fib(24) = 46,368
We know from the definition of the Fibonacci sequence:fib(n)=fib(n−1)+fib(n−2)fib(n) = fib(n-1) + fib(n-2)fib(n)=fib(n−1)+fib(n−2)
So:fib(24)=fib(23)+fib(22)fib(24) = fib(23) + fib(22)fib(24)=fib(23)+fib(22)
Plug in the known values:46,368=fib(23)+17,71146,368 = fib(23) + 17,71146,368=fib(23)+17,711
Now, solve for fib(23):fib(23)=46,368−17,711=28,657fib(23) = 46,368 – 17,711 = \boxed{28,657}fib(23)=46,368−17,711=28,657
Explanation
The Fibonacci sequence is a famous number pattern where each term is the sum of the two preceding ones. The sequence starts with:fib(0)=0,fib(1)=1fib(0) = 0, \quad fib(1) = 1fib(0)=0,fib(1)=1
Then each term is formed as:fib(n)=fib(n−1)+fib(n−2)fib(n) = fib(n-1) + fib(n-2)fib(n)=fib(n−1)+fib(n−2)
This recursive nature means that to find any Fibonacci number, we typically need the two numbers immediately before it.
In this problem, we are told that:
- The 22nd Fibonacci number is 17,711
- The 24th Fibonacci number is 46,368
We’re asked to find the 23rd Fibonacci number, fib(23).
Using the Fibonacci rule in reverse, we know that:fib(24)=fib(23)+fib(22)fib(24) = fib(23) + fib(22)fib(24)=fib(23)+fib(22)
This relation allows us to solve for the missing term:fib(23)=fib(24)−fib(22)fib(23) = fib(24) – fib(22)fib(23)=fib(24)−fib(22)
Substituting in the given values:fib(23)=46,368−17,711=28,657fib(23) = 46,368 – 17,711 = 28,657fib(23)=46,368−17,711=28,657
This approach works because the Fibonacci sequence is strictly additive. Each term is the sum of the previous two, so we can work forward or backward as long as we have two adjacent values.
This reverse logic is particularly useful in computational problems where some values are precomputed or known, and we need to fill in the missing ones. It’s also a fundamental concept in algorithms, dynamic programming, and mathematical proofs involving recurrence relations.
Therefore, the value of fib(23) is 28,657.
