Number Series Reasoning Tricks The Easy Way

in this video we're going to talk about how to find the next term in series of numbers so let's say if we have the numbers 2 and 14. what is the next number in the sequence now what we need to do is we need to look for pattern what pattern do you see in each of these numbers notice that each number has common difference of three to go from two to five you need to add three likewise to go from five to eight you need to add three five plus three is eight and eight plus three is eleven whenever you have sequence that differs by common number this is an arithmetic sequence if you can add or subtract by the same number you have an arithmetic sequence so 14 plus 3 will give us the next number 17. 17 plus 3 will give us the next number 20. so anytime you have sequence that differs by repeated addition or subtraction it's an arithmetic sequence now let's work on another example let's say if we have these numbers 5 9 13 17. what's the next number in the sequence find the next three numbers actually feel free to pause the video so notice that 5 differs from 9 by 4 units if we add four to five it will give us nine and if we add four to nine it will give us thirteen so therefore the common difference is four so to find the next number in the sequence we need to add 4 to 17 17 plus 4 is 21 21 plus 4 is 25 and 25 plus 4 is 29 here's different example 27 and then nine find the next three terms in the sequence so what is the common difference if we take the second number and subtract it by the first number we're going to get negative six if we take the third number and subtract it by the second number it will give us the same number negative six so therefore the common difference is negative six so if you add negative six to twenty seven you're going to get 21 and if you add another negative 6 to 21 it will give you 15. or if you subtract 21 by 6 you get 15. 15 minus 6 is 9. 9 minus 6 will give us the next number 3. and then 3 minus 6 will give us negative and then negative 3 minus 6 will give us the last number negative 9. so the common difference is negative since once you have that then you could find the next term in the sequence so once again this is another arithmetic sequence or arithmetic sequence here's different problem to try let's say if we have the sequence 3 6 12 and 24 what is the next number in the sequence so first let's see if there's common difference if we take the second term and subtract it by the first term it will give us difference of three and if we take the third term and subtract it by the second we're gonna get difference of six and so that's not going to help as much but let's divide the second term by the first term six divided by three is two and if we take the third term divided by the second we will get two so therefore this sequence has common ratio of two notice that to get from three to six or to go from three to six we need to multiply by two three times two is six and if we multiply six by two that will give us twelve and then if we multiply twelve by two it will give us 24. so this is what's known as geometric sequence because the numbers they differ by multiplication or division as opposed to addition and subtraction so to find the next number in the sequence we got to multiply 24 by 2 which will give us 48 and then if we want to find the next one it's going to be 48 times 2 which is 96 and so forth here's another similar example let's say if we have the numbers 4 12 36 what is the next number in the sequence so let's see if it's an arithmetic sequence or geometric sequence let's see if there's common difference if we take the second term and subtract it by the first we'll get eight and if we take the third and subtract it by the second this will give us 24. so therefore we don't have an arithmetic sequence let's test to see if it's geometric sequence 12 divided by 4 will give us 3 and 36 divided by 12 is 3. so we have common ratio if we multiply 4 by 3 it will give us 12. and if we multiply 12 by 3 that will give us 36 so to find the next number we got to multiply 36 by 3 and that's going to give us 108 and if we want to find the next two numbers we need to continue to multiply by three 108 times three is 324 and 324 times three is 972 and so that's it for this example go ahead and find the missing numbers in the sequence so feel free to pause the video to find the answer if we look at the first example one squared is the same as one two squared is four three squared is nine 4 squared or 4 times 4 that's 16. so the next number has to be 5 square which is 25 and then it's 6 squared which is 36 and 7 squared which is so look out for exponents because sometimes you might have that as pattern try this one 64. and then go ahead and find the missing numbers now eight is the same as two to the third two times two times two is eight twenty-seven is three to the third four to the third or four times four times four that's 64. therefore the next number in the sequence has to be five to the third which is 125 and then we have six cubed which is 216. and the next one will be 7 to the third and then so forth find the pattern consider the sequence 4 6 13 and 18. so go ahead and find the next numbers well we're not dealing with exponents here because 6 13 and 18 those are not perfect squares or perfect cubes so let's see if there's common difference or common ratio now if we subtract six by four that will give us two and if we divide six by four that's one point five if we subtract nine by six that will give us three and if we divide nine by six that's also one point five if we take 13 and subtract it by 9 that's 4. and if we take 13 and divided by 9 it's no longer 1.5 it's actually 1.4 repeating now if we take 18 and subtracted by that gives us 5 and if we take 18 and divided by 13 that's 1.3846 now notice that with division there is no pattern that we can use here but with subtraction there is pattern even though we don't have common difference the difference is increasing by one so to go from four to six we need to add two and to go from six to nine we need to add three to go from nine to thirteen we gotta add four and to go to 13 to we need to add 5. so therefore the next number has to be 18 plus 6 which is 24. and then after that we need to add 7 24 plus 7 is 31 and then we need to add 8. 31 plus 8 is so based on the pattern that we see here that's how we could find the next set of numbers here's another example using those five numbers find the next three numbers so what is the difference between seven and nine if we take nine and subtract it by seven we're going to get difference of two if we take 13 and subtracted by nine we're going to get difference of 4. 19 minus 13 is going to be difference of 6 and 27 minus 19 is difference of 8. so therefore in order to find the next number we can follow the sequence here the sequence of addition increases by two so next time we gotta add 10 27 plus 10 is 37 and then we need to add 12. 37 plus 12 is 49 and then we'll add 14. 49 plus 14 is 63 and so that's it that's how you could find the next three numbers sometimes you need to see the pattern in fraction so let's say if we have these four fractions three over four five over seven seven over ten and 9 over 13. so based on this find the next three fractions so how do we do this how would you begin so take minute and see if you can figure this out now when dealing with fractions personally find it helpful to separate the numerator and the denominator if we focus on the top numbers we could see pattern between 3 5 7 and 9. the common difference between those numbers is two three plus two is five five plus two is seven seven plus two is nine so therefore the next three top numbers have to be 11 13 and 15. now let's focus on the denominator 4 7 10 13 notice that each of those numbers differ by 3. four plus three is seven seven plus three is ten ten plus three is thirteen so thirteen plus three is sixteen and then we have nineteen and then nineteen plus three is twenty-two and so you want to separate the fractions into the top and the bottom part portion just see it separately and it's can help you to figure out the missing terms here's another example eleven over four eight divided by nine five over sixteen based on those three numbers find the next two numbers in this sequence so if we focus on the top three numbers eleven eight and five notice that the common difference is negative three if we subtract 11 by 3 it's going to give us 8. 8 minus 3 is 5. so 5 minus 3 is 2 and 2 minus 3 is negative 1. so that's going to satisfy the pattern on top on the bottom what pattern do you see 4 9 16 they're all perfect squares 4 is 2 squared 9 is 3 squared 16 is 4 squared so the next numbers have to be 5 squared and 6 squared 5 squared is 25 6 squared is 36. and so the missing numbers are 2 over 25 and negative 1 over 36. so that's it for this video so now hopefully you understand how to find the next term in sequence just by looking at the patterns by the way for those of you who want to have access to my pre-algebra video playlist take look at the description section i've left the link there so you can check that out when you get chance you