Hi I need help with the question below, I need to use arithmetic sequences to solve the question! "Mandy's friend Hahn deposited $200 in a bank on the 1st of January and on the first day of each of the following months. At the end of June, he found that his account balance was $1206.30. Find the rate of simple interest paid by the bank"

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Let's find the month interest rate `m,` then the year (simple) interest rate will be `12m.`

There are `6` full months elapsed till the end of June. The entire amount deposited is `6*$200=$1200.` So the profit is `$6.30.`

At the first month the profit was `m*$200.` At the second, `(2m)*$200` ($200 was deposited two times). Then `(3m)*$200` and so on, up to `(6m)*$200` (including). This is an arithmetic sequence, each term is $200 greater than the previous.

We want to find the sum of an arithmetic sequence with `6` terms, whose first term is `200m` and the last is `6*200m.`

The formula is `S_n=n(a_1+a_n)/2,`

here it is `6*(200m+6*200m)/2=200m*21($).`

And this is the profit, it is equal to `$6.30.` So `200m=0.3,` `m=0.0015.` It is the same as `0.15%` (monthly). Yearly it is `12*0.15%=1.8%.`


The answer: the annual rate of simple interest is 1.8%.

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