 Hello and welcome to another problem-solving session on sum of arithmetic progressions. Let's solve this given problem The sum of first six terms of an AP is 42. That's one information which is given then the ratio of its 10th term and 30th term so first of all there is a ratio 10th and 30th term is 1 is 2 3 find the first and 13th term So let's keep an eye on what exactly we have to find out first 13th term so we'll start with that only So solution So we have to find out a1 and a13 So it's always good to find out or write the objective In the first statement first step itself now what's given let's see Sum of first six terms of an AP is 42. So in terms we can write s6 is 42 Correct a and d are not known. So let the let the first term d a and common difference b d Right So I'm writing it common difference in abbreviated form whenever you are writing in an exam in a formal exam Please write full words So common difference bd Okay, so s6 can be written as by our formula. We know s6 is 6 by 2 that is n by 2 Times 2 times first term that is a plus 6 minus 1 d. This is 42 Correct this information is given as 42. So that means 3 into 2 a plus What 6d Or 5d rather 5d 6d not not 6d 5d 2 a plus 5d Is equal to 42, right? So you can cancel this 3 out and you will get 2 a plus 5d is equal to 14 This is equation number one Let's write it as equation number One now second thing is given as a 10 ratio of 8 10 to a 30 See it's given right ratio of 10th to 30th term is 1 is 2 3 So what this will be it will be 1 by 3 Okay, now what is a 10 a 10 will be simply n 10 by 2 So let me write the formula for sorry a 10 it is so a 10 is first term First term plus 10 minus 1d. So 90 is a 10 And this one a plus 20 90 is 30th term Is 1 by 3 So this is the knowledge of previous sessions that a n is equal to a first term plus n minus 1d Right now cross multiply and solve 3 a plus 27d is equal to a plus 29d Right So what will you get you will get 2 a is equal to 2d That means a is equal to d another equation You got a is equal to d Okay, so if a is equal to d from 1 From 1 and 2 What can you say you can say? 2 a plus 5 a because d is a is 14 So 7 a is 14. So a is equal to 2 Very easy a is equal to 2 What was the objective of the question to find first term a 13 So first term is now known. So a 1 is equal to 2 first is done And what will be a 13 a 13 will be a Plus 12d Now a is equal to 2 is equal to d because a and d were same. See so hence a 13 will be equal to 2 plus 12 into Correct So this is nothing but 26. So we got a 13 as well Right. So what was the learning learning is Some of first six terms of nap So you have to learn the formula sn Is n by 2 application of this formula was there Then the ratio of two terms were given and ratio of two terms means you need to know a n formula a plus n minus 1b these two things only were applied here And with the given constraint or whatever condition was given we could solve it and found out if a 1 and a 13