Math Is Fun Forum

teo

Member

Registered: 2007-08-04

Posts: 7

sequences and series

I put $k in a bank savings account on the first day of each month. At the end of each month, interest is added to the account at a fixed rate of 0.5% of the total amount in the account. Assuming that i did not withdraw any money from the account, show that at the end of n months, i have

$(1.005k + 1.005^2k + 1.005^3k+......+1.005^nk)

find the minimum number of months I have to save so that at the end of which the total amount of money in the bank exceeds $100k.

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to take the first step is easy, to climb up the mountain is another matter.

atavistic

Member

Registered: 2007-09-03

Posts: 2

Re: sequences and series

Easy.

remember the formula for compound interest??The formula is A=P(1+r/100)^t where A is total amount ,p is principal amount,r is rate of interest and t is time.So for this question A for the first month will be A=k(1+0.5/100)^1 = k(100.5/100) = k(1.005).

Similarly for second month it will be A= k(1+0.5/100)^2 = k(1.005)^2 and so on.

If u notice we get a geometric progression if we keep on adding.Therefore at the end of nth month money in bank =

(1.005k +1.005^2k + 1.005^3k +......+1.005^nk) = k(1.005 + 1.005^2 + 1.005^3......+1.005^n)

Now for the no of months when the account has 100k will be when the sum of terms inside the bracket above is 100.So applying sum of GP formula we get :

1.005(1.005^n-1)/1.005-1 = 1.005(1.005^n-1)/0.005 = 201(1.005^n -1 ) = 201 * 1.005^n -201

Now this value should be equal to 100.So

201*1.005^n -201 = 100
1.005^n = 301/201

1.005^n = 1.4975

From here u will get N to be approximately 80 months.SO thats the anwer