Download Harmonic and Fibonacci Sequences

Acc. Geom/Adv. Algebra
Name _______________________
Harmonic and Fibonacci Sequences
Period ______Date_____________
Definition: A harmonic sequence is a sequence in which all of the terms are reciprocals
of the terms of an arithmetic sequence.
Example #1: Since 2, 4, 6, 8, … is arithmetic, 1/2, 1/4, 1/6, 1/8, is harmonic.
Example #2: Since an = 4n  7 is arithmetic, hn = 1/(4n  7) is harmonic.
What type of function should a harmonic sequence graph?_____________________
Why?
Sometimes it is easier to recognize a harmonic sequence if you create common
NUMERATORS for your numbers. For example, consider the sequence: 6, 3, 2, ….
There is not a common difference so it is not ________________, and there is not a
common ratio, so it is not _________________.* However, the common multiple of all
these numbers is 6. So try writing all these terms with a numerator of 6: 6/1, 6/2,
6/3,… Now the harmonic sequence is clearer and we can write the nth term as tn = 6/n.
*Three non-collinear points will also create a quadratic function, so this sequence could also be quadratic.
To insert k harmonic means between numbers a and b, insert k arithmetic means
between 1/a and 1/b, then write the reciprocals of those means. OR write the
numbers with a common numerator and find the arithmetic means of the denominators.
Exercises:
1.) Write an explicit formula to find the nth term of 67/3, 67/7, 67/11, …__________
Use that to find the 25th term.
2.) Prove that 6, 8, 12, 24, … form a harmonic sequence.
Find the nth term.
3.) Given the numbers, 4 and 8:
A) Insert an arithmetic mean between the two numbers._______________
B) Insert a geometric mean between the two numbers. ________________
C) Insert a harmonic mean between the two numbers._________________
D) Write these three means in order from lowest to highest.
4.) Given the positive numbers, a and b, write the arithmetic mean, the geometric
mean and the harmonic mean and decide if the order (from lowest to highest) still
applies, and why it does.
Continued on the back
The Fibonacci sequence is the sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, …
What is the next term of this sequence? _______________
Write a recursive formula for this sequence.
1.) Take any three sequential Fibonacci numbers. Multiply the outer two together and
square the middle one. What do you notice?
2.) Take any four sequential Fibonacci numbers. Multiply the outer two and multiply the
inner two. What do you notice?
3.) Create a new sequence by finding fn/fn +1. Does this sequence converge or diverge?
If it converges, what number appears to be the limit?
4.) If n is a prime number, what appears to be true of fn ?
5.) Find two consecutive Fibonacci numbers, fn and fn +1. Then locate f2n +1 .
What do you notice about these numbers?
6.) Classify the following sequences as arithmetic, geometric, harmonic, Fibonacci,
or other. Then write the nth term (explicitly if possible).
A) 1, i, 1, i, …
B) log 2, log 4, log 8, log 16, …
C) 4/3, 4/5, 4/7, 4/9, …
D) 1, 1, 2, 3, 5, 8, …
E) 12, 6, 4, 3, …
F) 3  i, 3, 3 + i, 3 + 2i, …
*G) 4, 5, 9, 14, 23, 37, 60, …
H) 6/23, 13/23, 20/23, 27/23, …
I) 3, 4, 6, 9, 13, 18, …
For more interesting facts about the Fibonacci sequence, check out:
http://www.mathsisfun.com/numbers/fibonacci-sequence.html or
https://math.temple.edu/~reich/Fib/fibo.html