solving recurrence relations to solve a and b

Question

I am asked

A department store offers a budget of account to its customers. Each month interest is charged on any outstanding debt, while a fixed sum has to be repaid at the end of each month. A customer had an outstanding debt on 3 consecutive months of £750, £690 and £628.80. Find the interest rate and the fixed sum repaid each month.

So I know ${u_1 = 750, u_2 = 690, u_3 = 628.80}$

and using ${u_{n+1} = au_n + b}$

I can get 2 equations

${u_2 = u_1 + b}$

=> ${690 = 750a + b}$

And

=> ${u_3 = u_2 + b}$

=> ${628.50 = 690a + b}$

If I times the second by - 1 I can add the equations together

${750a + b = 690}$

${690a + b = 62.8}$

which makes a = 1.02

But according to the book, the interest rate is 2%.

Answer 1

$a=1.02$ represents the fact that the balance at the end of the month, before the payment is applied, is $1.02=102\%$ times the balance at the end of the previous month. The $100\%$ is the old balance, the $2\%$ is the interest.