 Hello and welcome to the session. In this session we will discuss the following questions that says, A sinking fund was created by setting aside $100 at the end of the first year and then at the end of each of the following years an amount 10% more than that set aside at the end of the immediate previous year. Find the total amount of the fund at the end of 25 years if the rate of interest is 8% per annum. Now, in case of immediate annuity, the first payments will start at the end of the first period and so the first installment would be equal to A into 1 plus I to the power of n minus 1. The first installment will earn interest for n minus 1 periods. In the same way, the second installment would be given as A into 1 plus I to the power of n minus 2. The second installment will be paid at the end of the second period and it would earn interest for n minus 2 periods and so on. We can find the payments of the third, fourth and other installments. Now, the last installment would be equal to A only as it would not earn any interest since its payment is done at the end of the term. Now, in this case this A that is in these cases A is the annual payment of each installment then n is the number of periods r percent is the rate of interest I is equal to r upon 100. This is the key idea that we use in this question. Let's now proceed with the solution. Let's see what all is given in the question. It's given that a sinking fund is created by setting aside $100 at the end of the first year and then $100 is set aside at the end of the next years and we have to find the total amount of the fund at the end of 25 years and we are also given the rate of interest as 8% per annum. So, according to the question we are given the annual payment of each installment A as $100 that is this is the amount which is set aside at the end of the first year. So, A is $100. Now, n would be equal to 25. Since we have to find the total amount of fund at the end of 25 years so n is equal to 25 then r percent is 8% I would be equal to r upon 100 that is equal to 8 upon 100 which is equal to 0.08. This is the value of I. Now, as in the question we have that this amount that is $100 is set aside at the end of the given period so this means that it is the case of immediate annuity and so from the key idea we find that in case of immediate annuity the first installment is given as A into 1 plus I to the power of n minus 1. So, here we have first installment would be equal to 100 that is A into 1 plus I which is 0.08 this whole to the power of n minus 1 so n is 25 and so n minus 1 would be 24 so here we have 24 that is the first installment will amount to 100 into 1.08 to the power of $24 and this installment will earn interest for 24 periods so first installment of $100 would be this at the end of 25 years now let's find out the second installment in the question we have given that at the end of the following years that is years following the first year an amount of 10% more than that set aside at the end of the immediate previous year is set aside so this means at the end of the second year 10% more than $100 which was set aside at the end of the first year would be set aside for the second year and so this would be equal to 100 plus 10 upon 100 into 100 and so this would be equal to $110 this would be small a for the second installment and the second installment is given as A into 1 plus I to the power of n minus 2 so second installment would be equal to 110 into 1 plus I which is 0.08 to the power of n minus 2 that is 25 minus 2 that is 23 so the second installment will amount to this which is 110 into 1.08 to the power of $23 at the end of 24 years this could also be written as 11 upon 10 into 100 into 1.08 to the power of $23 in the same way we can find out the third installment for which small a would be 10% more than 11 upon 10 into $100 so this would be equal to 11 upon 10 into 100 plus 10 upon 100 into 11 upon 10 into 100 so further we have 11 square is A for the third installment so third installment is equal to 11 square into 1 plus 0.08 to the power of 25 minus 3 which is 22 this is further written as 11 square into 1.08 to the power of 22 or we can also write this as 11 upon 10 whole square into 100 into 1.08 to the power of 22 so the third installment amounts to this at the end of 23 years so we can find out the next installment in the same way now the last installment is the amount A only as this amount will not earn any interest and it is paid at the end of the given time if you observe the pattern we find that the second installment is equal to 11 upon 10 to the power of 1 into 100 into 1.08 to the power of 23 the third installment is 11 upon 10 whole square into 100 into 1.08 to the power of 22 that is when finding the second installment we put here 1 while finding the third installment we put here 2 and so when we have to find out the last installment we will put 24 here that is 25 minus 1 which is 24 into 100 so 11 upon 10 whole to the power of 24 into 100 dollars is the last installment now we have to find the total amount of the fund at the end of 25 years so for the total amount we would add the values of all these installments so the first installment which is 100 into 1.08 to the power of 24 dollars so we write here 100 into 1.08 to the power of 24 plus the second installment which is 11 upon 10 to the power of 1 into 100 into 1.08 to the power of 23 plus the third installment which would be 11 upon 10 this whole to the power of 2 into 100 into 1.08 to the power of 22 plus and so on plus the last installment which is 11 upon 10 whole to the power of 24 into 100 now next the total amount would be equal to we will take 100 into 1.08 to the power of 24 common and this would be multiplied by 1 plus 11 upon 10 to the power of 1 into 1 upon 1.08 plus 11 upon 10 this whole to the power of 2 into 1 upon 1.08 to the power of 2 plus and so on plus 11 upon 10 this whole to the power of 24 into 1 upon 1.08 to the power of 24 we hold so this is further equal to 100 into 1.08 to the power of 24 this into 1 plus 11 upon now when we multiply this 10 with 1.08 we get 10.8 plus 11 square upon now 10 square multiplied by 1.08 whole square gives us 10.8 whole square plus and so on plus 11 to the power of 24 upon now 10 to the power of 24 multiplied by 1.08 to the power of 24 gives us 10.8 to the power of 24 the whole so this is further equal to 100 into 1.08 to the power of 24 into 1 plus 11 upon 10.8 plus 11 upon 10.8 the whole square plus and so on plus 11 upon 10.8 this whole to the power of 24 the whole as we can observe that this is a GP of 25 terms so we can find out the sum of this GP now sum of the GP of n terms is equal to a into r to the power of n minus 1 the whole this will upon r minus 1 when r is greater than 1 now in this case we have r that is the common ratio is equal to 11 upon 10.8 which is obviously greater than 1 so we can use this formula to find out the sum of this GP so using this we get this is equal to 100 into 1.08 to the power of 24 into a that is 1 into 11 upon 10.8 to the power of 25 that is the total number of terms is 25 in this GP minus 1 this whole upon 11 upon 10.8 minus 1 the whole this further gives us 100 into 1.08 to the power of 24 into 11 upon 10.8 to the power of 25 minus 1 and this whole upon 10.8 that is 11 minus 10.8 is 0.2 this upon 10.8 the whole further this is equal to 100 into 1.08 to the power of 24 into 10.8 upon 0.2 into 11 upon 10.8 to the power of 25 minus 1 the whole now we will find out the value of 11 upon 10.8 to the power of 25 by taking this to be equal to x now taking log on both sides we get log x is equal to 25 into log of 11 upon 10.8 so this is equal to 25 into log 11 minus log 10.8 the whole now putting the values of the logs of the given numbers we have 25 into 1.0414 minus the value of log 10.8 which is 1.0334 so this gives us 25 into 1.0414 minus 1.0334 which is 0.008 and so this comes out to be equal to 0.2 that is we have log x equal to 0.2 and from here we have x is equal to empty log of 0.2 which is equal to 1.585 this is the value of x which is 11 upon 10.8 whole to the power of 25 so this is equal to 100 into 1.08 to the power of 24 into 10.8 upon 0.2 into 1.585 minus 1 the whole which is equal to 100 into 1.08 to the power of 24 into 10.8 upon 0.2 into 0.585 now removing the decimals now to 54 times is 108 and so now further we have 54 into 58.5 into 1.08 to the power of 24 let us now find the value of 1.08 to the power of 24 using log so for this we suppose y equal to 1.08 to the power of 24 taking log on both sides we have log y is equal to 24 into log of 1.08 now further we get log y is equal to 24 into log of 1.08 is 0.0334 so now this gives us log y equal to 0.8016 and so here we have y is equal to anti log of 0.8016 thus y is equal to 6.333 which is the anti log of 0.8016 so 1.08 to the power of 24 is 6.333 so putting the value here we have 54 into 58.5 into 6.333 and this is equal to 2000 5 0.947 thus we have the total amount is equal to 20000 5 0.95 dollars or this could be equal to 20000 6 to the nearest dollar so the total amount is $20000 to the nearest dollar so this is our final answer this completes the session where we have understood the solution of this question