 Hello and welcome to the session. In this session we discuss the following question which says, Round purchase the house paying $220,000 and promises to make annual payment of $1500 each at the rate of interest 8% per annum compounded quarterly. If he misses first three payments, how much should he pay at the time of fourth payment to bring himself up to date? If suppose an annuity is left unpaid for N years, then the equivalent amount to be paid at the end of NH year is the same as the amount of ordinary annuity. So it would be given by small a upon i into 1 plus i to the power of n minus 1 d whole. This is capital A which is the amount that is to be paid at the end of the NH year if the annuity is left unpaid for N years. Where the small a is the annual payment of each installment, then N is the number of periods of the annuity. R% is the rate of interest per period i is equal to R upon 100 which is the interest of $1 for the same period. This is the key idea that we use in this question. So in the question we are given that Ram purchase the house paying $220,000 and he makes annual payment of $1500 each as the rate of interest is 8% per annum compounded quarterly. It's also given that Ram misses his first three payments and so we will have to find out the amount that he has to pay at the time of the fourth payment. So let's see the solution now. In this case we are given small a that is the annual payment of each installment as $1500. R% that is the rate of interest is 8% per annum compounded quarterly. So this is equal to 8 upon 4 which is equal to 2% per quarter and now i would be equal to R upon 100 that is equal to 2 upon 100 which is equal to 0.02. So here Ram misses the first three installments. Now he will have to pay the amount of the first three missed installments along with the installment of the fourth period also. Thus we can say that the total payment at the end of the fourth period would be the amount of annuity for a period. Hence the required amount to be paid would be equal to a upon i into 1 plus i to the power of n minus 1 the whole and in this case n would be equal to 4. Now putting the respective values we get this is equal to 1500 upon 0.02 into 1 plus 0.02 the whole to the power of 4 minus 1 the whole which is equal to 1500 upon 2. Now we remove this decimal and here we have in 200 this into 1.02 to the power of 4 minus 1 the whole 750 times is 1500 so we have this is equal to 7500 into 1.02 to the power of 4 minus 1 the whole. Let's now find out the value of this for this we suppose let x be equal to 1.02 to the power of 4 taking log on both sides we have log x is equal to 4 into log of 1.02. So further we have log x is equal to 4 into the value of log 1.02 is 0.0086 which is equal to 0.0344 and from here we have the value of x as the empty log of 0.0344 which gives the value of x as 1.082 so here we have 75000 into 1.082 minus 1 the whole further we get 75000 into 0.082 Now removing this decimal we have here 1000 in the denominator these three 0 cancel with these three 0 and we have 75 into 82 which is equal to 6150 So the amount to be paid at the time of fourth payment is equal to 6150 dollars so this is our final answer this completes the session hope you have understood the solution of this question.