is to find the sum of the first n term of a given arithmetic sequence so an example example number one we have find the sum of all the terms of the sequence 5 10 15 20 and so on up to 50. okay by solution you will be using listing all the terms of the sequence ebik savin mulasa first term hang gang's last term so that's why we have here myron thai on 5 plus 10 plus 15 plus 20 plus 25 plus 30 plus 35 plus 40 plus 45 plus 50 which is the last term okay adding all the terms using calculator the answer is 275 again terms thus the sum of the terms of the sequence is equivalent to 275 is the formula we have here the formula s sub n equals n over 2 times 2 times a sub 1 plus d times n minus 1 such that sub n the sum of the first n terms so the sum of the first and terms and a sub one gangnam pattern is the first term d is the common difference okay let us have example find the sum of the first 20 natural numbers again you are going to find the sum of the first 20 natural numbers and natural example one two three four and so on um let's say a sub 1 or the first term anina sabinat and one and pinaka so therefore a e sub 1 or the first term is 1. a sub n is the last term so our last term is 20 and n is the number of terms so from 1 to 20 there are 20 numbers and now it's a subnet and i sum of all the terms from 1 to 20 which is s sub n since the last term is given we use the following formula s of n is equal to n over 2 times a sub 1 plus a sub n again given an last term attack a first term so ito nothing formula okay by substitution a sub 20 is equal to 20 which is the value of n over 2 times 1 the first term plus 20 which is the last term so adding 1 and 20 then dividing 20 divided by 2 will give you s sub 20 is equal to 10 times 21 and 10 times 21 is equal to 210 so therefore the sum of the first 20 natural numbers is 210. okay let's have another example example number two find the sum of the first sixteenth term of the arithmetic sequence eight eleven fourteen seventeen and twenty so we have only one two three four five terms is the formula so again let us know or ala mean and given so 8 is the first term or a sub one then sixth and term is the number of terms and to get the difference or the common difference is the subtract molang 11 minus 8 or 14 minus 11 at 17 minus 14 and result on i d is equal to 3 so now you're going to solve for s sub 16 y 16 since there are 16 terms to be totaled by solution the gametime formula so our first term is given but the last term is not given decathlon first term is given and last term given formula last term so that's why it own formula we have s sub n equals n over 2 times 2 a sub 1 plus n minus 1 times d of a okay by substituting the given values in the formula is okay so we have here is 16 is equal to 16 the number of terms divided by 2 times 2 and a sub 1 is equal to 8 plus n again 16 minus 1 times the common difference is 3 okay mult dividing 16 divided by 2 will give you 8 2 times 8 equals sixteen and sixteen minus one equals fifteen bring down three then multiply first fifteen times 3 we have 45 then copy the remaining adding 16 plus 14 will give you 61 then copy eight and now with economic multiply number eight times 61 the answer is 488 so the value of s16 is equal to 488 thus the sum of the first 16 terms of the series is 480.