 Hello and welcome to the session. In this session we will discuss the following questions that save. At the beginning of each quarter, $500 are deposited into a savings account that pays interest at the rate of 12% per annum compounded quarterly. Find the balance in the account at the end of 4 years. Now according to the question the amount is deposited into the savings account at the beginning of each quarter. So this is the case of annuity due. So the amount of annuity due given by A is equal to small a upon I into 1 plus I the whole this whole into 1 plus I whole to the power of n minus 1 the whole. Where this a is the amount of annuity due and the small a is the annual payment each installment small n is the number of periods of annuity then r percent is the rate of interest per period then I which is the interest of $1 for the same period is equal to r upon 100. This is the key idea that we use in this question. Let's now move on to the solution. So in the question it's given that at the beginning of each quarter $500 are deposited and the rate of interest is 12% per annum compounded quarterly and we have to find the balance in the account at the end of 4 years. So this balance would be calculated using this formula for the amount of annuity due. Now we will consider this formula for the amount of annuity due because the payments are made at the beginning of each quarter. So this is the case of annuity due and so this formula would be used. So now let us see what is A which is the annual payment of each installment. According to the question it's $500 so small a is equal to $500. Now the time given is equal to 4 years n which is the number of periods of annuity and also the interest is compounded quarterly so n would be the number of quarters in 4 years which would be equal to 4 into 4 that is 16 so n is equal to 16. Now r percent is given as 12% per annum compounded quarterly so r percent is equal to 12% per annum and so 12 upon 4 which is equal to 3% quarterly. So i would be equal to r upon 100 that is equal to 3 upon 100 which is equal to 0.03. Now the amount of annuity due is given by a 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. Now putting the respective values we get a is equal to small a that is 500 upon i which is 0.03 into 1 plus i that is 1 plus 0.03 the whole and this whole into 1 plus 0.03 which is the value of i this whole to the power of 16 that is n minus 1 the whole. This further gives us a is equal to 500 upon 0.03 into 1.03 this whole into 1.03 to the power of 16 minus 1 the whole. Now the specimen cancels with this decimal so we now have a is equal to 500 into 103 upon 3 into 1.03 to the power of 16 minus 1 the whole. We will now calculate 1.03 to the power of 16 for this we assume let x be equal to 1.03 to the power of 16. So taking log on both sides we have log x is equal to 16 into log of 1.03 from here we get log x is equal to 16 into the value of this log is 0.0128 which gives us log x equal to 0.2048 so x would be equal to the empty log of 0.2048 which is equal to 1.603 this is x that is 1.03 to the power of 16 is 1.603 so now we have a is equal to 500 into 103 that is 103 upon 3 into 1.603 minus 1 the whole which now gives us a equal to 500 into 103 upon 3 into 0.603 now we remove this decimal so here we have upon 1000 these two zero cancels with these two zero and five two times is 10 and 3 201 times is 603 so we get a is equal to 103 into 201 upon 2 which is equal to 20703 upon 2 which gives us 10351.50 does we have a is equal to 10351.50 dollars hence balance in the account at the end of four years is 10351.50 dollars so this is our final answer this completes the session hope you understood the solution of this question