Hi, I need help with this number pattern involving fractions...?
952627
My answers are:\na_6 = (6^2+1)/2(23) = 37/46 ~= 0.709\na_7 = (7^2+1)/2(29) = 50/58 ~= 0.860\n(even though a_6 seems to violate the monotonic increasing)\n\nWrite a_i = n_i / d_i.\nI think I found the pattern for the numerator.\nn_i seems to be (i^2+1):\na_0=1/2 = (0^2+1)/2(1)\na_1=1/2 = (1^2+1)/2(2)\na_2=5/8 = (2^2+1)/2(3)\na_3=10/14 = (3^2+1)/2(7)\na_4=17/22 = (4^2+1)/2(11)\na_5=13/16 = 26/32 = (5^2+1)/2(16)\n\nThe denominator d_i is more tricky, it doesn't have a simple difference relation:\nd_5 = 2(16) = d_4 + 2(5)\nd_4 = 2(11) = d_3 + 2(4)\nd_3 = 2(7) = d_2 + 2(4) ??\nd_2 = 2(3) = d_1 + 2(1) ??\nd_1 = 2(2) = d_0 + 2(1) ??\nd_0 = 2(1)\n\nSo I think I found this pattern:\nd_5 = 2(5*3+1)\nd_4 = 2(4*3-1)\nd_3 = 2(3*2+1)\nd_2 = 2(2*2-1)\nd_1 = 2(2*1) \nd_0 = 2(1*0+1) = 2(1)\nwhich kind of has a pattern:\nd_7 = 2(7*4+1) = 2(29)\nd_6 = 2(6*4-1) = 2(23)\n\n__________________________________________________\nMy previous tries:\n\nTricky. Couldn't spot any simple algebraic or difference relation in the terms a_i.\nHowever you can see the terms are in increasing order:\na_0=a_1= 1/2 = 8/16 = 0.5\na_2= 5/8 = 10/16 = 0.625\na_3=5/7 ~= 0.714\na_4=17/22 ~= 0.773\na_5=13/16 = 0.8125\n\nSo for a_6, a_7 , we want to construct plausible answers > 13/16\n\nLooks like the terms seem to be converging on 1, or something close to 1, with a decreasing step, it might be easier to spot the pattern this way:\na_0=a_1 = (1 - 1/2)\na_2 = (1 - 3/8)\na_3 = (1 - 2/7)\na_4 = (1 - 5/22)\na_5 = (1 - 3/16)\nor a pattern looking at the steps:\n(a_2 - a_1) = 0.125\n(a_3 - a_2) ~= 0.89\n(a_4 - a_3) ~= 0.69\n(a_5 - a_4) ~= 0.38\n\nMaybe these are the closest rational approximations to (1 - k^n) with denominators <32, for some real k.\n\nSo if we look at the differences b_i = (1 - a_i )\nb_0=b_1= 1/2 = 8/16 = 0.5\nb_2= 3/8 = 6/16 = 0.375\nb_3=2/7 ~= 0.286\nb_4=5/22 ~= 0.227\nb_5=3/16 = 0.1875\n\nThis looks maybe more useful, you can see:\nb_2 = b_1 * 3/4\nb_5 = b_2 * 1/2
704151
Alright, first write them as improper fractions.\n15/8 and 7/4\n15/8 and 14/8\n30/16 and 28/16\nhow does 29/16 sound?
1102373
i wont answer.... log onto www.mathguru.com and discover solutions with explanations.
887265
Please double check the output because there is no pattern.  Also add the level of Math this is
529333
Hi\n\nYou start by trying to divide 13 by 60 (i.e. how many times does 60 go into 13?) The answer of course is - it doesn't. So you know your answer must be a decimal number. Next add a zero to the 13 making it 130. Now try again. How many times does 60 go into 130. The answer is two with 10 left over. So the start of your answer is 0.2; Next add a zero to your leftover 10 making 100. How many times does 60 go into 100? Answer is 1 with 40 left over. So your answer is now at 0.21; next add a zero to your leftover 40 making 400. How many times does 60 go into 400. Answer is 6 with 40 leftover. So your answer has now moved on to 0.216; next add a zero to your leftover 40........ Can you see the pattern?? You just keep doing this until you want to stop :-) In this particular case the 6 will keep re-occurring and so your answer will never end; it will become 0.2166666666666 going on forever. Rounded this makes 0.22. Finally, add on the 34 degrees to the decimal and you get 34.22.\n\nNow you see why Clive Sinclair invented the modern calculator.
1102516
by any chance is this a question from your maths homework? lol :D\n\nI cant work it out myself because to put it simple, im crap at maths.
889816
1/rt3 *1/rt2-1/rt3\n[1/rt3](1/rt2-1)\n[1/rt3](1-rt2/rt2)\nrationalising\n[(1-rt2)*rt6]/6
292709
The first term is 7C0 d^7 (2e)^0\nThe second is 7C1 d^6 (2e)^1\nThe third is 7C2 d^5 (2e)^2\nThe fourth is 7C3 d^4 (2e)^3\nSee how the pattern works -\n\nThe fifth term is 7C4 d^3 (2e)^4\n\n7C4 = 35; (2e)^4 = 16e^4; 35 times 16 = 560\n\nSo it simplifies to 560 d^3 e^4.
90506
I'll take it as far as I can.\n\n3 - Add all the numbers and if the result is divisible by 3 then the entire number is 9+1+2+0+3+1+2=18 which is divisible by 3.\n4 - The last 2 numbers are divisible by 4. 12 is divisible by 4 so the entire number is.\n6 - It's divisible by 2 and 3.\n8 - If the last 3 numbers are divisible by 8, then the entire number is. 312/8=39. \n9 - Like for 3, except it has to be divisible by 9. The result, from above, is 18 which is divisible by 9.\n12 - If divisible by 3 and 4.\n18 - If divisible by 2 and 9.\n24 - If divisible by 8 and 3.\n36 - If divisible by 4 and 9.\n72 - If divisible by 8 and 9.
603285
this is possible to figure out.  What it is a combination of numbers   i will do examples to keep it simple.   If your number was 00  or 000  then the answer was 9.  well its a pattern    if your number is 001 then the answer he came up with was 8.  well then if you had 002  then it was 7.  but also on the note if your number was 011 or some combination of this number then it was 8 also  because when you take 011 and add 0+1+1 you get 2 so it was 8.   the way it works is the numbers you type in it adds them together so there is a correlating number that goes with the answer that was added together.  so... if the numer added together was 0 then it was 9.  and the number added together was 1 then he came up with 8.  added together you got 2 then he said 7.  and just keep going ... 3 is 6,   4 is 5.    also on the same account    5 is 4,  6 is 3,   7 is 2,   8 is 1,  and 9 is 0.
990473
Start with demonstrating the multiplication of fractions through physical models.\n\nSplit a rectangle in half.  That is 1 x 1/2... because you are counting that one thing 1/2 time.\n\nThen split the 1/2 into 1/2.  1/2, counted 1/2 time, is 1/4.  See it?\n\nOK, so... through whole numbers you have already demonstrated (and they should already have shown true comprehension of the idea) that division is the inverse of multiplication. So...\n\n\n1/2 divided by 2/3 is the same as 1/2 x 3/2 because that is the inverse. Division of whole numbers is a reduction of value... division of fractions is an increase of value.\n\nAlso, try this with decimals.\n\n1/10 divided by 4.\n\n1/10 divided by 4/1.\n\n1/10 x 1/4 = 1/40.\n\n1/40 = 0.025.\n\n   ____\n4 |0.1     = 0.025.\n\n25 x 4 = 100\n\n2.5 x 4 = 10.0\n\n.25 x 4 = 1.00\n\n.025 x 4 = 0.100\n\nStart to see the pattern now?
796326
I'll just tell you enough and then you do the rest.\n\nEach of them involves a number which is a mixture of whole number and fraction.  First step is to exress it as an improper fraction, e.g.\n2) 3 5/8 = 24/8 + 5/8\n             = 29/8  \n(simple rule is \n(denominator*whole number + numerator)/denominator)\nLikewise 2 1/2 = 5/2\n\nI expect you can do\n(1/2)*(29/8)*(5/2)\n\n3) Same idea for 4 1/5, only extra thing to know is that to divide by 14 you multiply by 1/14.\n\nCheers
984538
(2/6) / (4/9)\n1/3 * 9/4\n3/4
214242
notice that if a remainder occurs twice, then it will immediately repeat.  hence the period is at most 111.  Yes, the pattern is distinctive, thats because 111 = 1 + 10 + 100\ntry for example base 7 with 1/(111) where 111 is in base 7.
559152
are u sure its not supposed to be the other way around. Cuz that would make a little more sense.\n\nBut if not, then it would probably be:\n1 + ((4s squared - 11) / (s to the 4 - 3s squared -11))\n1 the quotient. the rest the remainder.
236201
It is very simple, first you should write it this way:\n\n1/3 2/4 3/5 4/6 5/7 , ...\n\nthe answer is n/(n+2) , n being a natural number.
1006693
6, 10, 11, 20, 24, 40, ...\nThe diff between 6 and 10 is 4\n10 & 11 = 1\n11 & 20 = 9\n20 & 24 = 4\nso it's 4, 1, 9, 4\n4 and 9 and 1 are perfect squares, so just follow the pattern!