 Hello friends welcome again to another problem-solving session The question says in an AP the sum of first n terms is 3n square by 2 Plus 5n by 2 so an expression for the sum of first n terms of an AP has been given You have to find out 25th term Okay, so expression for SN is given so What is given SN sum of first term as a function of n is given 3n square upon 2 plus 5n upon 2 correct and we have to find out a 25 now if you remember We had discussed while we were discussing the sum of n terms of an AP that In any AP SN minus SN minus 1 Is an this is one of the Results which we obtained while analyzing the sum of n terms of an AP correct. So we are going to use this Concept in this question Alternatively, you can also solve through the you know the direct method that is direct application of the formula. What is SN? SN is given as n upon 2 Twice a plus n minus 1d Right so from this also we can see let's see both the approaches, right? So the first approach is SN minus SN minus 1 is a n. So what is SN is given? What is SN minus 1? SN minus 1 will be wherever there is an n replace it by n minus 1 square Sorry n minus 1 simply so 3n minus 1 square by 2 plus 5n minus 1 by 2 This is SN minus 1, right? So we will get a n as SN minus SN minus 1 so I'll write the expressions So it is 3 by 2 n square plus 5 by 2 What is that n and minus 3 by 2 n minus 1 squared plus 5 by 2 n minus 1 Now you can club n n square together or 3 by 2 together and what will be left with you is n square minus n minus 1 whole square correct and this one is plus 5 by 2 n and minus n minus 1 Right, so let's now simplify this further So 3 by 2 n square minus n minus 1 whole square is n plus n minus 1 within the identity and We can write this as n minus n plus 1 so a minus v a plus b form and This one is plus 5 by 2 n and n will get cancelled. We will be left only with a 1 Okay, so this is 3 upon 2 Now within brackets, what is it 2 n minus 1? 2 n minus 1 and this is simply one so okay only this much plus 5 by 2 Right, so let's do it further. So this is 3 n. Let's open the bracket 3 into 2 n by 2 is 3 n and Then minus 3 by 2 plus 5 by 2 Right, so this becomes 3 n and What will this be 3 n? 5 by 2 minus 3 by 2 is 1 So 3 n plus 1 correct so we got a n a n as 3 n plus 1 right so a 25 we have to find out a 25 so what is a 25 is it this was a n so a n is equal to this much so a 25 will be 3 times 25 plus 1 76 Now let's try to solve this problem Through another approach through the formula method right so sn is given to be n by 2 2 a plus n minus 1 d So we can find out s1 very easily right so s1 is how much? so put n equals to 1 so it will be 1 by 2 and and It will be 2 times a So in this case if you expand this you will get a n a plus a n square by 2d and Minus n by 2d correct and Arranging this in order you will get n square d by 2 Okay, and this one will be n common and it is a minus d by 2 right, so and sn is given as n square times 3 by 2 plus n times 5 by 2 So if you compare this one, this is through the formula and this is through given right so you can compare D by 2 will be equal to 3 by 2 the coefficient of n square that means d is equal to 3 Okay, and from the other side a minus d by 2 will be equal to 5 by 2 if you compare the coefficient This coefficient and this coefficient of n so this implies a is equal to d plus 5 by 2 Which is nothing but 3 plus 5 by 2 which is 4 so you got a s4 Correct, so you can now find out a 25 a 25 is A plus 24d right, so that means a is 3 plus 24 into Sorry a was 4 so a is 4 a is 4 Plus 24 into 3 which is again 76 so in both the cases You will get the value as 76 right, so there are two approaches of solving this problem