 Hello and welcome to the session. In this session, we discussed the following question that says, finally these number of years for which an annuity of $1200 per annum interest run in order that its amount just exceeds $25,000 at 8% compounded annuity. Now, in case of immediate annuity, we have A is equal to small a upon i into 1 plus i to the power of n minus 1 the whole, where this property A is the amount of annuity small a is the annual payment of each installment then n is the number of periods annuity is the rate of interest per period and this i is equal to r upon 100. This is the key idea that we use in this question. In the question, we are asked to find the least number of years that is we have to find m and we are given that the annuity is of $100 per annum the amount is also given to us and also the rate of interest is also given to us. Let's proceed with the solution now annuity runs for to find m period. So, this is the case of immediate annuity and so we will use this formula. The question we have the amount of annuity as $25,000 plus annuity given by capital A is equal to $25,000 and small a which is the annual payment of each installment is given as $1200. So, small a is equal to $1200 then the rate of interest per period that is r percent is given as 8 percent per annum and so i is equal to r upon 100 that is 8 upon 100 equal to 0.08. Let us now consider the formula capital A is equal to small a upon i into 1 plus i to the power of n minus 1 the whole where we have to find out the m. So, putting the respective values we have 25,000 is equal to 1200 upon 0.08 into 1 plus 0.08 that is i this whole to the power of n minus 1 the whole. Now, moving this decimal we put here 2 zeros. So, we now have 25,000 into 8 upon 1200 is equal to 1.08 to the power of n minus 1. Now, these 3 zeros cancel with these 3 zeros and 8 15 times is 120, now 5 3 times is 15 and 5 5 times is 25. So, we have 5 upon 3 is equal to 1.08 to the power of n minus 1 and from here we have 1.08 to the power of n is equal to 5 upon 3 plus 1 which is equal to 8 upon 3. And this is equal to we now have 1.08 to the power of n is equal to 2.666. Now, taking log on both the sides we get n into log of 1.08 is equal to log of 2.666. And thus from here we have n is equal to log of 1.08. Now, log of 2.6 0.4259 upon log of 1.08 is 0.0334. This will cancel with each other and so we have n is equal to 4259 upon 334 and thus we have n is equal to 12.75. Hence, the least number of years is equal to 13 years. That is 12.75 is ordered off to 13 and so this is our final answer. This completes the session. Hope you have understood the solution of this question.