Can you figure out the Fibonacci  Sequence?
351788
The Fibonacci series has a number of interesting properties.\n\nThe ratio of successive terms (term divided by next term) converges on 0.618 (the Golden Ratio). Get a calculator and fill in the gaps below\n\n5/8 = 0.625\n8/13 =\n13/21 =\n21/34 =\n34/55 =\n55/89 =\n\nAND\n\nThe ratio of successive terms (term divided by previous term) converges to 1.618 = the inverse of the Golden Ratio.\n\n8/5 = 1.6\n13/8 = 1.625\n21/13 =\n34/21 =\n55/34 =\n89/55 =\n\nThe numbers appear in nature eg in the arrangement of leaves on a stalk, sunflower heads, pine cones etc.\n\nThe generating formula, each term is the sum of the last two, can be used to generate other series e.g.\n\n1, 3, 4. 7, 11, 18, 29, 47, 76,\n\nCompare this to\n\n1, 2, 3, 5, 8, 13, 21, 34, 55\n\nand notice how the difference between 55 and 65 is 21, the differemce between 34 and 47 is 13, the difference between 21 and 29 is 8 etc.\n\nThe series starting 1, 4 and 1, 5 and 1, 6  show similar properties\n\n1, 6, 7, 13, 20, 33, 53, 86, 139\n1, 5, 6, 11, 17. 28, 45, 73, 118\n1, 4, 5, 9. 14, 23, 37, 60, 97, \n1, 3, 4. 7, 11, 18, 29, 47, 76,\n1. 2. 3, 5, 8, 13, 21, 34, 55\n\nand they all converge on the golden ratio too.\n\nOne can experiment with other generating formulae\n\neg each term = the sum of the 3 previous terms\n\n0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81\n\nor each term is twice the previous term plus the one prior to that\n\n0, 1, 2, 5, 12, 29, 70, 169\n\nand explore the properties of such series. Have fun!
437
Well, many people answered the question. Fibonacci sequence starts from 0 and Nth Fibonacci Number is sum of (N-1) th and (N-2) th Fibonacci Number.\nSequence goes like 0,1,1,2,3,5,8,13,21,34…....\n\nIt is an empirical fact that the ratio of consecutive Fibonacci Numbers converges to an interesting value near to 1.6 called (to be accurate it is (1+SQRT (5))/2) golden ratio or golden mean.\nThere ere are some interesting facts about golden ratio:\n1.\nThe growth of nautilus shell, growth of populations and many other kinds of natural “growing,” are somehow governed by mathematical properties exhibited in the Fibonacci sequence. Not just the rate of growth, but the pattern of growth. Examine the crisscrossing spiral seed pattern in the head of a sunflower, for instance, and you will discover that the number of spirals in each direction is invariably two consecutive Fibonacci numbers.\n2. \nGolden ratio is related to construction of great ancient architectures like Great pyramids in Egypt. \nLink: http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html\n3.\nYou can follow this link for exploring its use in poetry, music, art and now a days it is used for stock market predictions as well. This link will tell you the origin of the word Fibonacci as well.\nLink: http://www2.evansville.edu/studiochalkboard/cm-fibonacci.html
399866
There is an easy way to solve it. \n\nThese numbers are known as "pyramidal numbers" as each number in the sequence is equal to the number of blocks in an equilateral pyramid. As a side note, each sum in brackets is known as a "triangular number", as it is equal to the number of blocks in an equilateral triangle (note - a pyramid is a stack of triangles).\n\nMy memory is a little fuzzy on these things, but I believe that the closed-form equation is this:\n\nP(n) = n * (n + 1) * (n + 2) / 6
572436
Arithmetic sequence:  1, 2, 3, 4, 5, 6, 7\nGeometric sequence:  1, 2, 4, 8, 16, 32, 64
902613
I just plugged this into my calculator.\n\n[1, 0, 2, 0, 2, 0 | 0\n0, 1, -3, 0, -2, 0 | 0\n0, 0, 0, 1, -1, 0 | 0\n0, 0, 0, 0, 0, 1 | 0\n0, 0, 0, 0, 0, 0 | 1 ]\n\nTake it however you want.
517585
Starting with 0, the sequence is the square of that number:\n\nf(n) = n^2. \n\nf(0) = 0 (0^2)\nf(1) = 1 (1^2)\nf(2) = 4 (2^2)\nf(3) = 9 (3^2)\nf(4) = 16 (4^2)\nf(5) = 25 (5^2)\netc.\n\nNote, it isn't clear if your indexes are going 0,1,2,3,4,... or are going in descending order 0,-1,-2,-3,-4... the resulting sequence would be the same.\n\nf(0) = 0 (0^2)\nf(-1) = 1 (-1^2)\nf(-2) = 4 (-2^2)\nf(-3) = 9 (-3^2)\nf(-4) = 16 (-4^2)\nf(-5) = 25 (-5^2)\netc.
1173688
You'll find plenty of interesting and relevant info. from this link. Just scroll down to Article links.\n\nhttp://www.wholesalecentralair.com/newsletters/Fibonacci_series.htm
1124750
No\n\n\n\nf(3+1)=f(4)=2(4)+1=9\n\nbut\n\nf(3) + f(1)= [2(3)+1] + [2(1)+1] = 7 + 3 = 10
383876
In an arithmetic sequence, the formula for the nth term is\n\ninitial term + (common difference times (n-1))\n\nIn other words, you start with a number and keep adding the same number to that original number. In the sequence you gave (the odd numbers) the first number is 1 and you keeping adding 2. (The reason why the nth term is 1 + (n-1)2 is that you started adding 2 with the second term.)\n\nso the 101st term of that sequence would be\n\n1 + (100)2 = 201\n\nThe sum of the sequence, also known as an arithmetic series, has its  own formula:\n\nSum = (initial term + last term)*n/2\n\nThis comes out of the fact that if you add the first and last term, the result will be the same as adding the second and second-to-last term, which will be the same as adding the third and the third-to-last term, etc.\n\nSo for 20 terms of the sequence above, the first term is 1, the 20th term is 1+(2)19=39 and the sum of the sequence is (1+39)*20/2=400.\n\nI'll leave it up to you to do the sum of the first 30 terms.\n\nHope that helps.
723065
The sequence is the given series is *2 *3 *4 *5 *6and so on...so the answer is 720
833353
this sequence is Hexadecimal. it has letter values for A, B, C, D and so on.\n\na Hexadecimal Number is based on the number 16. the Hexadecimal numbers look the same as the decimal numbers up to 9, but then there are the letters ("A',"B","C","D","E","F") in place of the decimal numbers 10 to 15.\n\nA = 10.\nBB = 187.\nCCC = 3276.\nDDDD = 56797.\n\nso 56797 is the next number in the sequence.
619316
Do you have the same homework as Tilo B?\n\nStarting with 0, the sequence is the square of that number:\n\nf(n) = n^2. \n\nf(0) = 0 (0^2)\nf(1) = 1 (1^2)\nf(2) = 4 (2^2)\nf(3) = 9 (3^2)\nf(4) = 16 (4^2)\nf(5) = 25 (5^2)\netc.\n\nNote, it isn't clear if your indexes are going 0,1,2,3,4,... or are going in descending order 0,-1,-2,-3,-4... the resulting sequence would be the same.\n\nf(0) = 0 (0^2)\nf(-1) = 1 (-1^2)\nf(-2) = 4 (-2^2)\nf(-3) = 9 (-3^2)\nf(-4) = 16 (-4^2)\nf(-5) = 25 (-5^2)\netc.
689300
Q98 = 1,1,2,3,5 \n\nP.S: Looks like the fibonacci series
1157976
0000, 0001, 0002, 0003, 0004, 0005, 0006, 0007, 0008, 0009, 0010, 0011... etc, etc  untill u get to 9998, 9999\n\ntry it out, might take a long time but one combination will work :))
726208
I think the easiest way to know whether or not you add is by picturing a number line that is negative on the left and positive on the right with zero in the middle.\nif you subtract, then you move left\nif you add, then you move right.\n\n\n\nMemorize this!\n+ + makes +\n+ - makes -\n- + makes -\n- - makes +\n\nexample:\n1+(+2)=1+2\n1+(-2)=1-2\n1-(+2)=1-2\n1-(-2)=1+2\n\nanother example:\n(-1) + (+2) = (-1) + 2 = 2 - 1\n(-1) + (-2) = (-1) - 2 = -(1+2)\n(-1) - (+2) = (-1) - 2 = -(1+2)\n(-1) - (-2) = (-1) + 2 = 2 - 1\n\ntry these, and we'll tell you if you got it\n4 - (-5) =____\n6 - 7 = _____\n-9 - 8 = ______\n-3 - (-5) = _______
315162
http://www.hepl.phys.nagoya-u.ac.jp/docs/pif500.text.gz\n\nHere's the website....I don't think I'll try putting it all down here.\n\nAnd here's the one for Phi\nhttp://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html#phi2000
1004221
For the moment we don´t even know if there is an infinite number of Fibonacci primes....\n\nHere are the first fifteen:\n\n2,3,5,13,89,233,1597,28657,\n514229,433494437, 2971215073, 99194853094755497,\n1066340417491710595814572169,\n19134702400093278081449423917,\n475420437734698220747368027166749382927701417016557193662268716376935476241\n\nLooks considerably less dense than the fraction prime numbers / natural numbers