Please explain the logic behind deriving a geometric sequence from compound interest.

Question

Consider this:

Ginny opens a savings account and decides to pay £200 into the account at the start of each month. At the end of each month, interest of 0.5% is paid into the account. Show that the total interest paid into the account over the first 12 months is £79.45 to the nearest penny.

Model 1 of solving used by the mark scheme:

1. After one month, the interest paid is 0.5% of £200, or £1.
2. After two months, the interest paid is 0.5% of £200 for the first month plus 0.5% of (£200 + £1) for the second month, or £1 + £1.005 = £2.005.
3. After three months, the interest paid is 0.5% of £200 for the first month plus 0.5% of (£200 + £1) for the second month plus 0.5% of (£200 + £1 + £2.005) for the third month, or £1 + £1.005 + £1.010025 = £3.015025.

This forms a sequence: 1, 1.0025, 1.0050125 with first term 1, and common ratio 1.005. From here the sum of the first 12 months can be calculated easily.

Alternative model 2 which makes more logical sense to myself:

1. Start of month 1= 200, end of month 1 = 200*1.005 = 201
2. Start of month 2=201+200=401, end of month 2 = 401*1.005=403.005
3. Start of month 3 = 403.005+200=603.005, end of month 3 = 603.005*1.005=606.020025.

Thus at the end of month 1, £1 is paid interest; end of month 2, £2.005 is paid interest; end month 3, £3.015025 is paid interest

Please can somebody explain the discrepancy behind this. If the model 1 (using geometric sequence) is correct then how would one derive this?

Thanks.

Answer 1

Model 1 should be:

1. After one month, the interest paid is 0.5% of £200, or £1.
2. After two months, the interest paid is 0.5% of £200 for the first month plus 0.5% of (£200 + £1) for the second month, or £1 + £1.005 = £2.005.
3. After three months, the interest paid is 0.5% of £200 for the first month plus 0.5% of (£200 + £1) for the second month plus 0.5% of (£200 + £1 + £1.005) for the third month, or £1 + £1.005 + £1.010025 = £3.015025.

So you get something like this for the interest using the Model 1 approach. You can also see the £606.020025 from model 2 (if you ignore the £600).

Start       Interest    Cumulative  Cumulative-cumulative         
200         1            1           1
201         1.005        2.005       3.005
202.005     1.010025     3.015025    6.020025
203.015025  1.015075125  4.03010013 10.05012513
204.0301001 1.020150501  5.05025063 15.10037575
205.0502506 1.025251253  6.07550188 21.17587763
206.0755019 1.030377509  7.10587939 28.28175702
207.1058794 1.035529397  8.14140879 36.42316580
208.1414088 1.040707044  9.18211583 45.60528163
209.1821158 1.045910579 10.22802641 55.83330804
210.2280264 1.051140132 11.27916654 67.11247458
211.2791665 1.056395833 12.33556237 79.44803695