How much should I deposit each month into a savings account which earns 6 per annum compounded monthly if I want to have 10000 immediately after the 20th deposit?

Answer 1

Let the amount deposited by a.

Each period the total amount is increased by r.

Initially there is an amount of a

After 1 period an extra a is deposited (2nd deposit) and the amount already there is increased giving a total amount of a + ra

After 2 periods an extra a is deposited (3rd deposit) and the amount already there is increased giving a total amount of a + r(a + ra) = a + ra + r²a

So after n periods and a total of n+1 deposits have been made the total amount is given by a + ra + r²a + ... + rⁿa

Let this sum be S; then:

S = a + ra + r²a + ... + rⁿa

rS = ra +r²a + r³a + ... + rⁿ⁺¹a

rS - S = rⁿ⁺¹a - a

→ S(r - 1) = a(rⁿ⁺¹ - 1)

→ a = S(r - 1)/(rⁿ⁺¹ - 1)

The apr is 6% → r = (1 + 6/100) for 1 year

As we want monthly, we need the 12th root of this, namely r = 1.06^(1/12)

As we want after the 20th deposit, n+1 = 20 → n = 19

In our problem, S = 10,000

→ a = 10,000 × (1.06^(1/12) - 1)/((1.06^(1/12))^20 - 1) ≈ 477.27 per month