00:01
Hello students, we are given in the question.
00:03
Find the semi -annual deposit necessary to accumulate $26 ,000 after 2 .5 years in account earning 5 .75 % compound interest semi -annually right.
00:17
So we all know future value in compound interest is principal value multiplied by 1 plus r divided by 100 raised to power t.
00:25
So definitely here compound here, compound interest, compound interest is semi -annual right is semi -annual what does it imply r -dice is equal to r divided by 2 means 5 .75 % divided by 2 right so we need to find that so this is 2 .5 years so total if i take total time will be 5 because t -daz would be 2 multiply by t it means 2 by 2 .5 so basically 5 is the total time period because so now let's make our equation let's say p is our principal value so p multiply by 1 plus r so for first year this would be something like this so for second year this is p multiply by 1 plus 0 .0575 divided by 2 raise to power 2 plus p multiply by 1 plus 0 .0 575 divided by 2 raise to power 3 plus p multiply by 1 plus 0 .0 575 divided by 2 raise to power 4 p multiply by 1 plus 0 .0 575 divided by 2 raise to power 5 and this total value is equivalent to $26 ,000 right so now what we are doing we need to write a particular instruction here rate is 5 .75 % semi -annually compounded so we need to let the semi -annual semi -annual deposit to be p right so definitely we can write here so every semi -annual deposit to be p so we can write here we do here that future value future value of five deposits will be five deposits if i write here is equivalent to is $26 ,000 right so basically if i write here what would be the value so we need to apply here using some of geometric progression series right geometric progression so we all know of a so if a plus ar plus ar square up to a r, it is a geometric series having a is first term and r is common ratio.
03:16
So this is equal to a multiply by a multiply by r raised to power n minus 1 divided by r minus 1.
03:25
So now we need to apply that.
03:27
So what would be the answer? so let's take.
03:30
So if i take this equation number 1 from equation 1, we can can write from if we have to extend this from equation number one we can write this k t multiply by p multiply by one point if i write here this will come out as so p multiply by and this is first term one plus point 0575 divided by 2 this whole term can be calculated as 1 .02875 multiply by 1 .02875 raise to power 5 minus 1...