7 On the first day of each month, Kate pays \(\pounds 50\) into a savings account. Interest is paid on the total amount in the account on the last day of each month.
The interest rate is 0.2\% At the end of the \(n\)th month, the total amount of money in Kate's savings account is \(\pounds T _ { n }\) Kate correctly calculates \(T _ { 1 }\) and \(T _ { 2 }\) as shown below: $$\begin{aligned} T _ { 1 } & = 50 \times 1.002 = 50.10
T _ { 2 } & = \left( T _ { 1 } + 50 \right) \times 1.002
& = ( ( 50 \times 1.002 ) + 50 ) \times 1.002
& = 50 \times 1.002 ^ { 2 } + 50 \times 1.002
& \approx 100.30 \end{aligned}$$ 7

1. Show that \(T _ { 3 }\) is given by $$T _ { 3 } = 50 \times 1.002 ^ { 3 } + 50 \times 1.002 ^ { 2 } + 50 \times 1.002$$ 7
2. Kate uses her method to correctly calculate how much money she can expect to have in her savings account at the end of 10 years. 7
3. 1. Find the amount of money Kate expects to have in her savings account at the end of 10 years.
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4. (ii) The amount of money in Kate's savings account at the end of 10 years may not be the amount she has correctly calculated. Explain why.