On 1 October 2015, one thousand euro was invested in EUROBANK at a fixed rate of

Question

On 1 October 2015, one thousand euro was invested in EUROBANK at a fixed rate of 3% per annum. This interest is compounded annually. No more lodgements are made. No withdrawals are to be made. Find: (a) a(1), the amount in the account after 1 year - i.e., on 1 October 2016; (b) a(2), the amount in the account after 2 years - i.e., on 1 October 2017; (c) A recursive formula for a(n), the amount in the account after n years. (d) A closed form solution for a (n), the amount in the account after n years. (e) The amount in the account on 1 October 2031.

Explanation / Answer

Let P = Amount invested = 1000 Euro

n = number of years

i = rate of interest per annum.

a) Amount in account after 1 year = P * (1+i)1 = 1000(1.03) = 1030 Euro

b) Amount in account after 2 year = P * (1+i)2 = 1000(1.03)2 = 1060.9 Euro

c) A recursive formula for a(n)

Let ao be amount invested now.

So after 1 year amount coumpounded to a1 = (1+i) ao = ao (1.03)

After 2 years

a(2) = (1+i) a1 = 1.03 a1 = 1.03(1.03 ao)

= (1.03)2 ao

For n years

a(n) = (1.03)n ao .

This is recursive formula for a(n)

d)

After n years amount is 1000*1.03n

e)

There are total 16 years up to 1 October 2031

So amount = 1000(1.03)16 = 1604.71 Euro.

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