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A Formula for the nth Term of the Fibonacci Sequence:

Updated on June 28, 2015


Paragraphs 13, 14, 15, 16, 17, 18 and 19 address whether or not the Apostle John has a problem with telling the truth. Paragraph 20 is concerned with my secretaries' retirement.

I have not put very many links in this hub because I've grouped these hubs concerning the derivation of this formula, and you will find this grouping at the very end, actually just past the end.

INTRODUCTION--OK, 2nd introduction:

2. It has been kind of a long journey to this point. Hub#12.5 put the FS into a matrix and matrix equation; #12.6 found the eigenvalues; #12.9 found the eigenvectors; #12.22 found the inverse matrix; #12.23 found the diagonal matrix. This hub will give us the trillionth term of the FS, and any other term you would like to calculate.


3. Remember P is the eigenvector matrix for the FS matrix( which is A ), and P-1 is the inverse of P; D is the diagonal representation matrix( DRM) of the FS matrix. The formula for D at hub#12.23 is repeated at L1. It can be proven that if A is raised to the nth power then each entry in D is raised to the nth power; therefore, if A is squared then the entries in D are squared also. This is what was done at the 2nd matrix equation at L1.

4. Remember that matrix multiplication is not commutative; i.e. AP ≠ PA, which is read matrix A times matrix P does not equal matrix P times matrix A. Under some conditions it can be true, but you do not want to do the matrix algebra depending on that assumption. With this in mind we want to isolate An by multiplying both sides of the equation at L1 with matrix P, but do it by approaching each side from the left. You can't do one side from the left and the other side from the right. Next multiply both sides of the equation with P-1 approaching both sides of the equation from the right. Notice at L2 P times P-1 cancel into the identity matrix. So we have the identity matrix times An times the identity matrix and we are left with just An . It is like multiplying any number by the number one: nothing changes. So An has been isolated at L2.


5. Now we just multiply matrices as we have been doing all along in this group of hubs. So L3 writes out the matrices to be multiplied, and L4 begins with the first two matrices, P and Dn . If you can not see the last matrix at L4, that is ok because it is repeated at the left side of L5, which is then multiplied by P-1 . In multiplying the matrices at L5 we are only interested the 1st element, a1,1 . Remember at Hub#12.5( linked) it was determined that the first element of the FS matrix represented the nth term. So if we raised the FS matrix, A, to the 7th power, then element a1,1 gave the 7th term of the FS. Therefore, and this is the really cool part, we do not need to multiply the matrices at L5; we only need to do the multiplying to determine the 1st element, a1,1 . This is what was done at L6. L7 is using the laws of math to simplify L6. L7 is the formula that we want, and L8 gives it in symbolic form. L8 reads: the nth term of the FS is equal to a1,1 .


6. I've explained the FS in a previous hub, and linked you to sites in that hub which explained it further. I listed the first 12 terms of the FS at L9. At L10 I factored out the common term to make the equation easier. We plug in 11 for F(n) = F(11) into the formula at L11 and we get 144 as the answer at L12, which is the correct answer. At L13 I plugged in 153 into the formula and got 6.83 X 1031 . This is the correct answer because I raised the FS matrix to the 153rd power with my Ti-89 calculator and got the same answer for a1,1 . At hub#12.5 we found the 4000th term of the FS by raising the FS matrix to the 4000th power with the TI-89 and got an answer of 6.46 X 10835. This is the same answer that our formula here gave us from L14 through L18.


7. My TI-89 can only go as high as 10999. It is many years old, and the newer ones may go higher. The way to handle numbers bigger( or smaller) than what your calculator can handle is to use logarithms. This is what is done for the remainder of this hub. This hub is not intended to explain that procedure; however, integralcalc( linked) has a helpful hub on the laws of logarithms. Paxwil( linked) takes it further. Cristina327(linked) has some interesting word problems solved with logs.


8. We want to determine the value of F(4000) at L14. Right off the bat my calculator could not do 22001 , because its value is 2.6364 X 101204 . We set F(4000) = y at L15, and then take the common log( base 10) to both sides of the equation. There is a lot of dividing and multiplying on the right side, which is perfectly suited for logarithms. Logs keep the numbers calmed down so that the calculator can work with them, and then you can exponentiate to get them back to normal( the way they were). The hubs I linked you to at paragraph 7 explain what I did at L15, L20, L21, L26, etc. It is just using the laws of logarithms to plow through the problem. At L16 we are ready to put the numbers back to "normal" so we exponentiate, which is a relatively new term. When I first studied logs it was called taking the antilog. The antilog is doing the reverse, which is placing 10( in this case) on both sides of the equation, and then raising the 10 to the power of the terms in the equation. The laws of exponents is used at L17, and since multiplication is commutative I switched them at L17 also. We have two factors there at L17; therefore, the first factor is 10 raised to the 0.810064 power,which gives 6.4575. The next factor is 10 raised to the 835 power. So the two factors put together is the number given at L18. This is the 4000th term of the FS( Fibonacci sequence).


9. This same procedure is followed for the remainder of these calculations in this hub.


10. The formula at L10 is not the same as at L14, L19 and L25. The reason why is the number (1 − √5)n+1 at L10 becomes insignificant as n becomes large, and eventually the accuracy of the calculator does not even include it. So why would we use it?


11. So at L23 the 1,000,000th term of the FS is listed, and at L28 the trillionth term is listed.


12. I think mathematics is fascinating, and it has been used to give us the technology we enjoy today, and it has made business enormously more efficient; however, It has been my experience that if you do not want to be bugged at parties then just start talking about the trillionth term of the Fibonacci sequence. They will all leave you alone. What's that? You're asking me if I can guarantee it? Yah I can pretty much guarantee it.


ED# 13.1 You can bet your eternal soul on that statement; nevertheless, some people erroneously believe that the original Scriptures have contradictions, inaccuracies and discrepancies. The following argumentation apparently gives--of all things--mathematical proof that these people are correct. Because it is "mathematical proof," I thought this would be a perfect example to prove the old saying, "Don't count your chickens before they hatch." When it is all said and done, John's statement was an understatement, not a overstatement.


13. Some people will argue that exaggeration is just another form of lying; therefore, is the Apostle John a liar? It appears so. He wrote at John 21:25, "And there are also many other things which Jesus did, the which, if they should be written every one, I suppose that even the world itself could not contain the books that should be written. Amen." The way I'm associated with numbers a good portion of the day, I guess it was inevitable I would check out John's statement eventually. Jesus lived for 33 years or 17,344,800 minutes. They did not have typewriters in John's time, but I'm in a generous mood so I'll let John have one. Furthermore, I'll give him professional secretaries who can each type 80 words per minute. Wikipedia, in their article, "Words Per Minute," said the word is standardized to 5 keystrokes including spaces; therefore, this works out to 400 keystrokes per minute( 80 X 5 = 400). Man alive! That is fast! That is nearly 7 keystrokes per second! I had to hire 4 full time secretaries and one part timer, who worked 8 hours per week plus filled in for vacation time for the other four during the 33 years Jesus Christ was alive. Then I had to transport the 5 secretaries back in time when Jesus was born. They had instructions to type everything Jesus said and done 24 hours a day, 7 days per week for 33 years. Our mission was to determine if the Apostle John exaggerates.


14. We know that Jesus slept( Matthew 8:24 and 25; Mark 4:48; Luke 8:23 and 24), but we want to be generous with John; therefore, we will assume that Jesus did not sleep from the time He was born. The secretaries took 8 hour shifts; therefore, typing 80 standardized words per minute they typed 6,937,920,000 keystrokes at the end of the 33 years, and it required 578,160 sheets of paper. It was the same kind of paper mentioned in my DNA, part 1 hub( linked); therefore, this stack of 578,160 sheets was 17.2 feet higher than our DNA stack, which made if 127.2 feet high, and having a volume of 82.59 cubic feet. This amounts to a cube that has a length of 4.355 feet to a side, but, again, lets be generous and make it 5 feet to a side. Amazing, isn't it? It does not look like the entire earth will be required, does it?


15. If memory is serving me correctly, when I wrote the Wormwood Letters it required 8 handwritten pages to equal 1 typed page. Being generous, let's say 10 pages; therefore, for a typewritten sheet( 2 pages) we have 20 handwritten pages( 10 sheets of paper). John did not have access to our paper technology so let's allow John's paper, or whatever they wrote on, to be 7 times thicker than our paper. These adjustments will make the volume 70 times bigger. This will give us a volume of 5,782 cubic feet or a cube of nearly 18 feet to a side. Ok, fine, let's round it up to 20 feet long, by 20 feet wide by 20 feet high. Looks to me that the earth can quite easily hold what could be written concerning what Jesus did.


16. It appears so. This is not just exaggeration, embellishment, or even lying; it flat out appears to be a completely ridiculous statement--absurd to the nth degree. How can anyone be this far off on their estimate? The Bible is God's inspired Word. Surely God knew that John's statement would look very foolish. How can this little bit of arithmetic not discredit John, discredit the Bible, discredit Christ, and discredit God?


17. Yah, I'm in trouble also. At hub#14( linked) I wrote some stuff that just does not seem to fit with my new discovery concerning John. At paragraph 6 I wrote that Christians "boldly stand by God's Word, and they unequivocally believe in its accuracy and inerrancy." At paragraph 2 I wrote, "those words in Scripture are God's words," and that God's prophets and servants were inspired by God to write what they wrote. The two Bible axioms( also at paragraph 2) will be disintegrated if this new knowledge can't be explained, but how can I possibly explain this? There is no manipulation of the math that will fix this! Amazing things can be done with math, but it can't change facts; it just confirms them. And for icing on this cake made from conundrum I wrote at paragraph 4 that God knows--and knew--every detail of the future. So it is not like God was caught off guard. He had, like, all of eternity past to prepare for the day that John wrote such an apparently ridiculous statement.


18. Now before you read this paragraph, stop and ponder what was said at paragraphs 13, 14, 15, 16 and 17. What does it do to your faith? If an atheist asked you about this, what would your answer be? Christians can not ignore facts as these. If we intend to lead others to Christ we need to supply solid answers, not superficial platitudes. We need to search the Scripture, and solve the problem. I admit it may take time. I was not able to come up with a reasonable answer concerning God's question to Job( paragraphs 5, 6, 7 and 8 of hub#14( see paragraph 17 above for the link)) until I learned of fractal geometry. Then the pieces of the puzzle fitted perfectly. God's Word is marvelously astounding, and unbelievably reliable. This "conundrum" presented in the first 31 words of the last verse of John's Gospel is answered in the first 31 words of his Gospel. I think that is amazing!


19. Believe it or not, I think this is the FACT we should be focused upon. I not only agree with John, but I'll go out on a Limb and boldly declare that if we hollowed out the earth and filled it with "the books that should be written" concerning the "many other things which Jesus did" that even then the "world[ earth] itself could not contain" them all. This bold statement for all Christians is proved here at paragraphs 12, 13, 14, 15, 16, 17 and 18 at this hub: Newtons-Method-a-Numerical-and-Iterative-Process-of-Finding-Difficult-Roots-of-a-Function.


20. I set my time machine to bring my secretaries back to me 7 minutes after they left. This gave me time to get a fresh cup of potent-hot coffee before they got back. Well, it is a time machine; I can do this. Of course in their time frame they were all 33 years older; therefore, when they got back they not only wanted their pay for 33 years of loyal work, but also a retirement program. They did not think much of my sense of humor when I said that I should only have to pay for 7 minutes each.


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    • Caleb DRC profile image

      Caleb DRC 4 years ago

      I have the same attitude, North Wind. I knew/know that John could not be exaggerating or embellishing but the math said, Yes he is! This is a perfect example of how people--especially atheists and apostate christians--jump on something like this to discredit God, the Apostles, the Bible, and the Truth. If we stand stalwart, and know the Bible is God's own inspired Word, then the answers to these "discrepancies" will eventually become clear. In this case the answer greatly glorifies God through Jesus Christ. I plan to begin giving that answer at the end of the next math hub.

      Well, yes. To make things worse those were mechanical typewriters--no electricity--so they had to be in good shape to type all day, everyday, on those. Ironically, that was not their complaint. Since they got back I have heard the end of, "We can't believe you sent us where we could not get our morning- Italian- coffee- lattes." They, once again, did not appreciate my sense of humor when I said, "Well, technically, I did not send you where there are no Lattes, I sent you when there are not coffee lattes."

    • North Wind profile image

      North Wind 4 years ago from The World (for now)

      Great way to get yourself alone at parties :) I could just picture it. I am looking forward to reading about John's statement some more. When I first read your analysis of it, I confess that I said to myself, 'I don't even need Caleb to do the Math, I know John was right.'

      Just wondering if those secretaries had mechanical hands because I do not know how someone could type that long without completely destroying their bone structure!

      Great hub as usual, Caleb.