 Hello and welcome to the session. In this session we will discuss the following question which says A scholarship of $21,000 per alum in perpetuity starting from end of six years from now is to be started by an institution It pays in four equal installments starting at the end of present year If the rate of interest is 7% find the amount of each installment it pays The present value per petrol annuity is given by A upon I also in case of immediate annuity The first payment 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 That is the first installment will earn interest for n minus 1 periods In the same way the second installment will amount to A into 1 plus I This will to the power of n minus 2 And the second installment is paid at the end of the second year And it earns interest for n minus 2 periods and so on that is in similar way the other installment could be found This is the key idea that we use in this question Let's now proceed with the solution In the question we are given that a scholarship of $21,000 per alum in perpetuity is to be started by an institution Such that the scholarship starts from the end of the sixth year And the institution pays in four equal installments starting at the end of the present year It's also given that the rate of interest is 7% We have to find the amount of each installment it pays Now the scholarship isn't perpetuity that means it is forever Suppose four equal installments of A are paid by the institution Starting at the end of the present year Now it's given that the scholarship would be starting from the end of the sixth year from now So this is our present year Now from this present year our sixth year is this So at the end of the sixth year the scholarship is started That is from here the scholarship of $21,000 is started And that is given in perpetuity Now we will have to find the amount that is required at the end of fifth year To give the scholarship from the end of the sixth year So to give the scholarship $21,000 from the end of onwards in perpetuity The amount required would be given by Using this formula that is the present value of perpetual annuity which is given by A upon I This would be equal to $21,000 upon I I is equal to R upon 100 Which would be equal to 7 upon 100 R upon 100 is equal to 0.07 So we write here 21,000 upon 0.07 We remove the decimal and now 7 3,000 times is 21,000 And so this is equal to $300,000 Now we can find out the four installments that the institution would pay And these would be found out by this formula The first installment would be equal to A Which is to be found out into 1 plus I That is 0.07 to the power of n minus 1 In this case would be 5 Installments are being made And these are being made at the end of these years So this is the case of immediate annuity And n would be considered as 5 So 5 minus 1 is 4 So we put here 4 So this is equal to A into 1.07 to the power of 4 Now in the same way The second installment would be equal to A into 1.07 to the power of n minus 2 That is 5 minus 2 which is 3 Now the third installment would be equal to A into 1.07 to the power of 5 minus 3 That is 2 In the same way The fourth installment is equal to A into 1.07 to the power of 1 Now these installments That is A into 1.07 to the power of 4 Plus A into 1.07 to the power of 3 Plus A into 1.07 to the power of 2 Plus A into 1.07 Would be equal to this amount That is 300,000 dollars Now taking A into 1.07 common This into 1.07 to the power of 3 Plus 1.07 to the power of 2 Plus 1.07 plus 1 The whole is equal to 300,000 So this is further Equal to A into 1.07 This whole into Now 1.07 the whole cube is 1.2250 Plus 1.07 whole square is 1.1449 Plus 1.07 plus 1 The whole is equal to 300,000 So we now have A into 1.07 into 4.4399 The whole is equal to 300,000 This gives us A into 4.7507 A is equal to 300,000 And from here we get A is equal to 300,000 upon 4.7507 So further we have A is equal to 63148.59 So we can say Hence the amount of installment Which needs to be deposited by the institution Is equal to 63149.00 Rounded off 10.00 This is our final answer This completes the question Hope you understood the solution of this question