 Welcome friends to another session on problem solving on arithmetic progression in this problem It's been given that the sum of fifth and ninth terms of an AP is 30 72 rather 72, right? And the sum of 7th and 12th terms is 97. These are the two Conditions given you have to find the AP and you know at the moment It is being asked to find the AP it means to find the first term and the common difference Sum of 5th and 9th terms of an AP is 72. So this is how you should proceed you assume the first term. So let's say let us say first term of The given AP given AP is equal to P1 Okay, and common difference Common difference. These are the two parameters of a of an AP Right very important parameters difference be D Right now What does the first condition says the first condition says T5 that is fifth term and T9 ninth term that sum is how much? 72 So you'll get one relationship from here another relationship is T7 Plus T12 is how much 97? So with these two conditions, you have to find out T1 and D. So let's try and find out what is T5 guys T5 is T1 plus how much? N minus 1 that is 5 minus 1 times D and T9 is T1 plus 9 minus 1 D and this is equal to 72 So simplifying you will say to T1 and this is 4d and 8d. So 12d is 72 Correct on the other hand if you see this is T7, which is T1 plus 7 minus 1 So I'm writing directly 6d and T12 is nothing but T1 plus 12 minus 1 that is 11d and this is equal to how much 97 So if you simplify this you'll get to T1 plus how much 11 plus 6 17d is equal to 97 Right now let us say this is equation number one Let us say this is equation number two and now let's do this exercise two minus one So if you do two minus one, what do I get? I get 2d plus 17d minus 2d minus 12d LHS and in the RHS you'll get 97 minus 72 so this one is nothing but 5d and This one is nothing but 25 So this will clearly give you D that is equal to 5, right? So you got one of the two parameters that is D and you just need to find out the you know a or the T1 So I'm now using this equation one to find out T1 So 2t1 plus 12 into D is 5 is 72 Okay, so 2t1 is nothing but 72 minus 60 is 12 Correct, so hence this implies D1 will be equal to nothing but 12 upon 2 which is 6 So you got the first term as well, right? No, so hence what will be the AP like so the given AP is Nothing but 6 first term then first term plus D D that is 6 plus 5 11 then 11 plus 5 16 then 21 then 26 So on and so forth. This is the given AP. This is what they have asked us to find out So what is the learning learning is if such questions are there clearly? This is a question of linear equation in two variables. You get two relation using the NF term formula of the AP Solve the two linear equations to get T1 and D So many questions would be like where there will be two equations in T1 and D or first term and the common difference You solve the equations and find out the two values and that's that is how you find the AP