growing annuity example future value formula excel

Question:You are considering whether your savings will be enough to meet your retirement needs.You saved 100,000 last year and you expect your annual savings to increase by 8 percent per year for the next 20 years.if your savings can be invested at 9 percent,how much you would have at the end of twentieth year?

Solution:In 0th year you will have 100000 with you.Given this amount is growing at 8 percent per year.So at the end of 1st year your amount will be 100000*(1.08).In second year it will 100000*(1.08)^2   

At the end of 20th year this amount will be 100000*(1.08)^20

Given that we are investing these amounts at 9 percent.

So the future value at the end of 20th year will be

Let 100000 be denoted by 1 lakh

Therefore future value

=1.09^20 +1.09^19*1.08+1.09^18*1.08^2+........................+ 1.08^19*1.09+ 1.08^20

This is a Geometric Progession with first term as 1.09^20 and G.R as 1.08/1.09

Therefore using the formula for a G.P with G.R<1

Sn=A*(1-GR^n)/(1-GR)

Where A is the first term which is =1.09

GR is Geometric ratio which is =1.08/1.09

n=21 as we have 21 terms in the sum equation

Therefore Sum=1.09^20*(1-(1.08/1.09)^21)/(1-(1.08/1.09))

                        =107.5 Lakh

 Please do comment below if you got any doubt.