 The arithmetic sequence ai is defined by the formula ai is equal to ai minus 1 plus 3, where the first term is given to us as negative 24. So this means that ith term in the sequence can be given by the sum of the term previous to the ith term and 3. For example, if we take i equals to 2, so the second term in the sequence can be given as sum of the first term and 3. Now the first term is already given to us as negative 24. So the second term would come out to be negative 24 plus 3, which is negative 21. Similarly, let's take another example. If i is equal to 3, that means the third term in the sequence can be given as sum of the second term and 3. And we just found out how much the second term is. This is equal to negative 21, so negative 21 plus 3 is negative 18. So using this, we can figure out the value of any term of the sequence. Now let's see what do we need to find out. And n if the sum of the first n terms is negative 105. So let's see. We are given an arithmetic sequence, we are given the sum of first n terms of the sequence and we need to figure out the number of terms in the sequence. Now as we have already seen in the previous videos, sum of the first n terms of an arithmetic sequence can be given by the average of the first and the nth term in the sequence times the number of terms in that sequence. So using this, let's try to figure out how many terms our sequence has. So let's think about what do we know in this equation and what do we need to find out in order to evaluate n. So we are already given the sum of the first n terms. So we know what sn is. The first term is given to us as negative 24. So we know about this. What we don't know is what is the nth term of the sequence. So let's try and figure out what an is. So to figure out an, let's quickly write our sequence. The first term is negative 24, then second is negative 21, negative 18 and this goes on till we get to our nth term. Now as we can see that between any two consecutive terms, the common difference is always 3. In other words, I can say to get to the second term, I am adding the common difference once to our first term. To get to my third term, I am adding this common difference twice to my first term. Similarly if we keep on following this pattern, to get to our nth term, I will have to add my common difference n minus 1 times to my first term. So now if you go ahead and replace this expression for an in this equation over here, you will get your old school formula which is something as sn is equal to n by 2 times 2a1 plus n minus 1d which is nothing but just another way to write this whole thing. So you can either do that or we can just simplify an over here. Now a1 is given to us as negative 24, n is something that we are still figuring out and d is 3. So let's simplify this expression, I will distribute 3 over n and negative 1. So this would become negative 24 plus 3n minus 3. And on simplifying this, we will get an equals to 3n minus 27. Now we have figured out the logic behind this problem. All you need to do is simplify this whole thing. So why don't you pause this video and give it a go. Now let's go ahead and substitute this value of an in this equation over here. So doing that sn is already given to us as negative 105, I will write that instead of sn and then a1 is negative 24, an is something that we just found out. I'll write 3n minus 27 as an and this whole thing divided by 2 times n. Now we have an expression in which there's just one variable n. So let's go ahead and simplify this whole thing. So in the numerator, negative 24 and negative 27 add up to negative 51. So I can rewrite this numerator as 3n minus 51. And also I can see a 3 common in both these terms. So probably I can pull that out too. So I'll write this term as 3 times n minus 17. Now on multiplying 2 on both sides of this equation, I'll have 105 times 2 on the left hand side and 3 times n minus 17 times n on the right hand side. So on dividing both sides of this equation by 3 on the left hand side, 3 goes into 105 exactly 35 times. So over here we have negative 35 times 2, which is negative 70. And on the right hand side, 3 goes into 3 once and this can be rewritten as n squared minus 17n. Looks like we are getting ourselves a quadratic equation. Alright, so let's add 70 on both sides of this equation and then we'll have n squared minus 17n plus 70 equals to 0. So let's try to find out what n should be. So let me just make some space on the screen. So we'll use factorization to solve this quadratic equation. I can write 70 as 10 times 7. So this equation can be rewritten as n squared minus 10n minus 7n. Now these two things add up to negative 17n and then we have plus 70 equals to 0. So we'll consider these two terms and these two terms at a time. So from this term we have, we can take n as common and we'll be left with n minus 10. And from this term over here we can take out 7 as common and we'll be left with n minus 10 again. So now in these two terms we have n minus 10 as common. So I can rewrite this as n minus 10 times n minus 7 is equal to 0. So this means either n minus 10 is 0 or n minus 7 is 0. So we can say n is either equal to 10 or n is equal to 7. So this means let me get my question back up here. Let me also delete some stuff from here so we can create some space. Now n comes out to be 7 and 10. Meaning if we take the sum of first seven terms of the sequence we'll get negative 105. Also if we take the sum of first ten terms of the sequence we'll again get negative 105. So how's this possible? This could only mean one thing that the sum of eighth, ninth and tenth terms should be 0. To understand what I'm saying let's just figure out our seventh term. So the seventh term can be found out using this expression. So seventh term would be 3 times 7 minus 27. So 3 times 7 is 21. So 21 minus 27 which is minus 6. So our seventh term is negative 6. So beginning with the seventh term let's figure out our eighth term. So I'll just write A7 below it and A8 our eighth term would be negative 6 plus 3 which is negative 3 then our ninth term would be negative 3 plus 3 which is 0 and the tenth term would be 0 plus 3 which is 3. So we can see from here that eighth, ninth and tenth term add up to a 0. So this is why we can say that either the sum of seven terms in the sequence is negative 105 or some of the first ten terms in this series is negative 105.