 Hello and welcome to the session. In this session we will discuss the following question that says, what is the amount of annuity due of $200 yearly payable half yearly for 10 years at 8% compound interest per annum payable half yearly. Before we move on to the solution, we will discuss the formula to find the future amount of annuity in case of annuity due. In case of annuity due, the amount of annuity given by a is 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 n is the number of periods of annuity. Then r% is the rate of interest per period and we find out i as r upon 100. This i is the interest of $1 for the same period. This is the key idea that we use in this question. Let's move on to the solution now. In the question we have to find out the amount of annuity due of $200 yearly payable half yearly for 10 years and the rate of interest is also given as 8% per annum payable half yearly. So this means we can use this formula to find out the amount of annuity due. First of all let us see what is the small a? That is the annual payment of each installment which is small a is equal to $200 divided by 2. We will divide it by 2 since the payment is done half yearly and so this is equal to $100 half yearly. Now let us see what is the time given to us. Time is now as the payment is done half yearly so n that is the number of periods of annuity would be equal to 2 into 10 that is equal to 20. Then again rate of interest that is r% is 8% per annum we divide this by 2 so this is equal to 4% half yearly. So now i would be equal to r upon 100 that is equal to 4 upon 100 equal to 0.04. Now the amount of annuity due which is given by capital A is 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. So we will put the respective values in this formula to get the amount of annuity due that is to find out the capital A. So we have capital A is equal to small a which is $100 half yearly so we write here 100 upon i that is 0.04 into 1 plus 0.04 the whole this whole into 1 plus 0.04 the whole to the power of n which is 20 minus 1 the whole. So here we have A is equal to 100 upon 0.04 into 1.04 into 1.04 to the power of 20 minus 1 the whole. Now this decimal cancels with this decimal and 4 26 times is 104 so we have A is equal to 2600 into 1.04 to the power of 20 minus 1 the whole. Now we will find the value of 1.04 to the power of 20 for this we suppose let X be equal to 1.04 to the power of 20. Now taking log on both sides we have log X is equal to 20 into log of 1.04 further we have log X is equal to 20 into. Now the value of log of 1.04 is 0.017 so now we have log X is equal to 20 into 0.017 which is 0.34. Now to find the value of X we take the empty log of 0.34 and this comes out to be equal to 2.188 this is the value of X that is 1.04 to the power of 20 is 2.188. Now putting this value here we have A is equal to 2600 into 2.188 minus 1 the whole so A is equal to 2600 into 1.188. Now removing this decimal we get here 1000 these to 0 cancel with these to 0 this is equal to 30888 upon 10 so we say A is equal to 3088.80 thus the required amount of sanity due is equal to 3088.80 dollars this is our final answer. This completes the session hope you have understood the solution of this question.