 Hello and welcome to the session. In this session we will discuss the following question which says a bank pays interest at the rate of 10% per annum compounded quarterly, find how much one should deposit in the bank at the beginning of each quarter in order to accumulate $5,000 in two years. Before we move on to the solution, let's discuss the formula to find out the amount of any t due, this is given by a and equal to small a upon i into 1 plus i the whole into 1 plus i the whole to the power of n minus 1 the whole. Where the small a is the annual payment of each installment, then n is the number of periods of annuity. Then we take r% as the rate of interest per period and so i which is the interest on $1 for the same period is equal to r upon 100. This is the key idea that we use in this question. Let's proceed with the solution now. In the equation we have that the rate of interest that is 10% per annum is compounded quarterly and we have to find the amount that one should deposit at the big link of each quarter so that $5,000 get accumulated in two years. Now as the amount would be deposited in the bank at the beginning of each quarter so this is the case of annuity due and so we will make use of this formula. Now as we have to find out that how much the person should deposit in the bank at the beginning of each quarter so this means we have to find the annual payment of each installment which is the small a. Now let's see what all is given to us. We are given that total amount accumulated in the bank is $5,000 so that is capital A is equal to $5,000. Time is given as 2 years n would be equal to 2 into 4 that is 8 since the rate of interest is compounded quarterly. Let's see what is our person given to us. It is 10% per annum so this is equal to 10 upon 4. Now 2 times is 4, 2 5 times is 10 so this is equal to 5 upon 2% quarterly since the rate of interest is compounded quarterly. So I would be equal to R upon 100 that is I would be equal to 5 upon 200 so the amount of annuity due would be given by capital A equal to small a upon I into I plus 1 the whole this whole into 1 plus I whole to the power of n minus 1 the whole. Now putting the respective values we have 5,000 that is capital A which is the amount of annuity due is equal to small a which we have to find out upon I. Now this I is 5 upon 200 which is equal to 1 upon 40 or you can say 0.025 is I so we write here 0.025 into 1 plus 0.025 that is the value of I into 1 plus 0.025 the whole to the power of 8 minus 1 the whole. Further we have 5,000 is equal to A upon 0.025 into 1.025 into 1.025 to the power of 8 minus 1 the whole. Now this decimal cancels with this decimal and so we have 5,000 into 25 upon 1025 is equal to A into 1.025 whole to the power of 8 minus 1 the whole. Let us now find out the value of 1.025 to the power of 8 for this we suppose let x be equal to 1.025 to the power of 8 now taking log on both sides we have log x is equal to 8 into log of 1.025. So further we have log x is equal to 8 into the log of 1.025 is 0.0107 so we now have log x is equal to 0.0856. Now we can find the value of x by taking the empty log of 0.0856 which comes out to be equal to 1.218 this is the value of x that is 1.025 whole to the power of 8 is equal to 1.218. So we now have 5,000 into 25 upon 1025 is equal to A into 1.218 minus 1 the whole. This gives us 5,000 into 25 upon 1025 into now 1.218 minus 1 is 0.218 so we write here 0.218 and this is equal to 8. We remove the decimal here and we put three zeros here now 25 41 times is 1,025 and 25,000 times is 25,000. Now 2 109 times is 218 and 2 500 times is 1,000 so we now have the small a is equal to 5,000 into 500 is 25 0000 upon 41 into 109 is 4,469. Now on dividing these two we get a is equal to 559.41 dollars so hence the required amount that should be deposited in the bank at the beginning each quarter is 559.41 dollars so this is our final answer just come these two sessions hope you have understood the solution of this question.