 Hello and welcome to another session on Selection of terms in an AP in this question. It's given that we have to divide 32 into four parts 32 into four parts 32 into four parts which are in AP, right? So the 32 has to be divided in four parts which are in AP such that product of extreme that is first in the fourth Will be the extreme terms is to product of means that means second and third Will be the means is 7 is to 15 Okay, so let's try and solve this problem now we have seen in the previous couple of sessions that whenever it's given that four Numbers are in AP then it's always advisable to take them in this form. So let's say a minus 3d a minus d a plus d and a plus 3d are Or in AP Okay, now, it's not mandatory to start with like this You can start with a a plus d a plus 2d a plus 3d But then we know that whenever the sum of the terms is given and which is clearly here Some is given because you have to break 32 into four parts That means if you add all the parts you will get 32 that means some is given so whenever some is given and AP terms are involved. It is always advisable to take terms in this Format rather than simply taking a a plus d a plus 2d and a plus 3d Why because it eases out a lot of calculations and you can illuminate one variable very easily now a minus 3d a minus d a plus d and a plus 3d are in AP and additional information is their sum is 32 right why some is 32 because 32 has been broken into four parts Obviously the sum has to be 32 Okay, so when you write this you have 32 here and See the you know the help you're getting now because now minus d and plus d will get cancelled plus 3d minus 3d will get cancelled You get 4 a is equal to 32 Hence clearly a is 8 now. It is much easier. Why because now you know one variable if you would have started with a a plus D, let's say if you would have started like this a plus a plus d Plus a plus 2d plus a plus 3d is Equal to 32 you would have got linear equation in two variables Which is also correct. It's not a problem, but then you know in this Step in this kind of in this kind of method We are getting an advantage that one of the variable d is eliminated and the value of the other is Obtained very easily. So a is 8. It makes our life much easier So an a is a let's go to the next Property which has been given or the you know information is given Product of extremes is to product of means now products of extremes are this Correct is to that means a ratio has to be taken product of means Okay, and this ratio is given to be equal to 7 times 15 7 by 15 not times 7 upon 15 So using the identities a plus be a minus be you can see this is a square minus 3d whole square and The denominator is a square minus d square and this up is 7 upon 15 So let's cross multiply a is anyways known so we can simplify, but let us simplify a little later So when you cross multiply you'll get 15 a squared minus 15 times 3d Squared is equal to 7 a square minus 7 times d square Okay, let's open the brackets and simplify 15 a square Minus 15 times 9 now 15 3 square is 9 and 15 times 9 is 135 135 d square is equal to 7 a square minus 7 d square This will reduce to 15 minus 7 is 8 a square and this becomes 135 minus 7 is 128 d square Okay, now this 8 is nothing, but this is 16 Okay, so hence d square is a square by 16 This implies d is plus minus a by 4 Right and a was 8 so hence plus minus 8 by 4 so hence plus minus 2 now deploying the value of one variable the very last step Make sure that you are you know, you're not doing any calculation errors it eliminates and you know It becomes a little easier right so you have just to calculate only once So d is plus minus 2 so hence what what happens we get a as 8 and D as either plus 2 and A is 8 and D is Minus 2 in both cases we can find the AP now first term was a minus 3d Right, so it is 8 minus 3 into 2 That is 2 here. It is 8 plus 3 into 2 that is Second is a minus d so 8 minus 2 so hence it is 6 Here it is 8 plus 2 hence it is then It is a plus d. So it is 8 plus 2 10. It will be 8 Minus 2 so it is 6 and finally a plus 3d is 8 plus 6 so 14 8 minus 6 so it is 2 Right, so in both the cases you'll see these are ascending order and descending order terms are same 2 6 10 14 Or 14 10 6 2 both if you see add up to 32 Right add up to 32 and you can check the sum of their squares Sorry, these are it's not the sum of the squares the product of their extreme that is 2 into 14 and product of their mean 6 into 10 is 7 upon 15 You can check that right So hence both the conditions are met and we have solved this problem. We have found out before for Numbers in AP. They are either 2 6 10 14 or 14 10 6 2