now let's say if you're given a sequence of numbers 2 4 6 8 10 and your goal is to write an equation that will help you to find any indicated term so what equation describes this pattern notice that we have an arithmetic sequence that increases by two the common difference is two so the equation is simply two n because it differs by two and you could test it if you plug in one two times one will give you the first term two if we plug in three two times three will give us the third term which is six if we plug in five 2 times 5 will give us the 10th term so when you plug in an n value you need to get the value of the term and then that's how you know if you have the right equation so based on that example what equation can we write that represents this pattern 3 6 9 12 15 and so forth so notice that the common difference is 3 so it's going to be 3n as opposed to 2n if you want to find the second term it's 3 times 2 which is 6. if we want to find the fourth term 3 times 4 it's 12. now what about this 1 3 5 7 9 and so forth we can see that it increases by 2. so there's going to be a 2n but if we plug in 2 into 2n we won't get 1 we're gonna get two since it increases as by a common difference i like to think of it as a linear equation here's the linear equation in slope intercept form the slope is basically the common difference because it increases by two the y-intercept it turns out that the y-intercept is the zero term the first term a of one is this one to find the term before it subtracted by 2 1 minus 2 is negative 1. so the equation is going to be 2x minus 1 or a sub n is 2 n minus one if we plug in one it's going to be two times one minus one which is one if we plug in four it's two times four minus one which is eight minus one that's seven that will give us the fourth term so it helps if you find a zero term before this one and that's going to be this number here consider this example 7 11 15 19. so based on a previous example write a general equation for this one so we could see that the slope or the common difference is four seven plus four is eleven eleven plus four is fifteen and so forth so there's gonna be a four n now the first term a of 1 is 7 what is the zero term so if we go back 4 units if we subtract 7 by 4 it's going to be 3 so that's like the y intercept that's the zero term so now let's find the fourth term based on this equation it's four times four plus three which is 16 plus three that's 19 and that will indeed give us the fourth term of the sequence now let's work on some other examples what about this one negative one one negative one one what equation can we write that represents this sequence if we have alternating signs this is simply negative one raised to the n power if you plug in one negative one to the first power is negative one if you plug in two negative one to the second power it's gonna be positive one negative one times negative one is one if you plug in three whenever you have an odd exponent it's going to be negative if you plug in 4 then it's going to be positive so now what if the order was reversed so what's the equation for this pattern of numbers all you need to do is either put a negative sign in front or what you could do is simply use n minus 1 instead of n both will work for example let's try to get the the fourth term using the first equation it's negative negative one to the fourth power negative one to the fourth power is positive one times a negative on the outside will give us negative 1. if we use the other equation negative 1 to the 4 minus 1 which is negative 1 to the third power because the exponent is odd overall it's going to be negative so both equations will work in that example now what about this 1 4 9 16 25 and so forth what pattern can you see here notice that these are all perfect squares the equation is n squared one squared is one two squared is four three squared is nine and so forth so now what about this what if we have alternating signs notice that all we need to do is combine the equation for this one and for this one because the first term is a negative the second is positive so a of n is going to be n squared times negative 1 raised to the n power here's the last example what if you have a sequence that is based on fractions how can you write a general equation that describes this pattern of numbers if you have a fraction what you want to do is write a separate sequence for the numerator and for the denominator of the fraction so for the top part of the fraction and numerator notice that the sequence of numbers is simply one two three four five so the general equation for that is simply n now for the numbers on the bottom they differ by 1 so that's 1n and if a of 1 is 2 we can see that a of zero the zero term is one less than two so it's one it's just n plus one if we plug in three three plus one is four will give us the third term the third term has the value of four so now we just got to put it together so a of m is equal to the numbers on top is n the numbers on the bottom is n plus one and so that's the general equation that describes uh that sequence so let's say if we have the sequence 7 10 13 16 and 19. the two things that you need is the first term and the common difference the first term is simply the first number which is seven the common difference you can find it by taking the difference of the second and the first term so in this case it's ten minus seven which is three you can also find it by taking the third term and subtracting it from the first i mean from the second term 13 minus 10 is 3 16 minus 13 is 3 and so the difference is the same between all numbers now once you have the common difference in the first term you can use this formula a of n is equal to a sub 1 plus n minus 1 times d so just plug in a 1 which is 7 and the common difference which is 3. and then simplify let's distribute the three to n minus one so it's three n minus three and then combine like terms seven minus three is four so therefore we have this equation it's three n plus four let's say if we want to find the fourth term just to check it a of four is three times four plus four three times four is twelve twelve plus four is sixteen so this method works so now it's your turn try this example let's say if we have the sequence negative 8 negative 6 negative 4 negative 2 and so forth write a general equation for the sequence of numbers using the formula so the first term is negative eight the common difference is the second term minus the first term which is negative six minus negative eight that's the same as negative six plus eight so the common difference is two so you have to add two to get the next term negative eight plus two is negative six negative six plus two is negative four and so forth so now let's use this formula to write the general equation so the first term is negative eight and the common difference is two so this is going to be negative 8 plus 2 n minus 2 and negative 8 minus 2 is negative 10 so this is the equation and let's check it the third term is going to be 2 times 3 minus 10. 2 times 3 is 6 6 minus 10 is negative 4 which is the third term of the sequence so let's say if we have the fourth term which is 39 and the seventh term which is 57 with this information how can we write the general equation in order to write the general equation we need to find the first term and the common difference now in the previous lesson we said that the fourth term and the third term differs by one common difference so the fourth term and the first term differs by three common differences because four minus one is three the seventh term and the fourth term also differs by three common differences since seven minus four is 3. let's use that equation to find d the seventh term is 57 the fourth term is 39 so let's subtract both sides by 39 57 minus 39 is eighteen and eighteen divided by three is six so the common difference is six now that we have the common difference we could find the first term the fourth term is thirty-nine and d is six three times six is eighteen and thirty nine minus eighteen is twenty one so the first term is 21 now that we have the first term and the common difference we can write the formula so it's a of n is equal to a of 1 plus n minus 1 times d so a of 1 is 21 d is six so let's distribute six so it's six n minus six and twenty one minus six is fifteen so this is the formula six n plus fifteen and we could check it let's calculate the seventh term so a of seven that's going to be six times seven plus fifteen six times seven is forty two forty two plus fifteen is fifty seven so it works you