Hello, Tangera!

Here's part of it . . .

A bank has an account for investors. Interest is added to the account

at the end of each year at a fixed rate of 5% of the amount in the account

at the beginning of that year. A man and woman both invest money

(a) The man decides to invest $x at the beginning of one year

and then a further $x at the beginning of each subsequent year.

He also decides that he will not draw any money out of the account.

i) Show that, at the end of n years, when the interest for the last year has

been added, he will have a total of: . dollars in his account.

At the start of year 1, he invests dollars.

At the end of year 1, it is worth: . dollars.

At the start of year 2, he invests another dollars ... total: dollars.

At the end of year 2, it is worth: . dollars.

At the start of year 3, he invests another dollars ... total: dollars.

At the end of year 3, it is worth: . dollars.

. . . and so on . . .

At the end of year , it is worth: . dollars.

. . The series has first term , common ratio , and terms.

. . Its sum is: .

Hence, we have: .

When is ?ii) After how many complete years will he have at least in his account?

We have: .

Take logs: .

Therefore: .