Question

Mrs. Speas has $138 in her bank account during Week #1. At the end of each week, she deposits $55 into her bank account.

What is the first term?

What is the difference?

What is the iterative rule for the amount of money Mrs. Speas has after n weeks? (you are NOT required to simplify)​

Answer 1

1) The first term is $138

2) The difference is $55

3) The iterative rule for the amount of money Mrs. Speas has after n weeks is 27.5·n² + 110.5·n

Explanation:

The mount Mrs. Speas has in her bank account = $138

The amount of money she deposits each week = $55

The amount of money in Mrs. Speas account therefore, forms an Arithmetic Progression

1) The first term = a = The initial money Mrs. Speas has in her bank account = $138

2) The (common) difference = d =The amount she deposits at the end of each week = $55

The iterative rule for the amount of money Mrs. Speas has (in her bank account) after n weeks is given by the formula for the sum of an arthmetic progression (AP), Sₙ, as follows;

`S_n = (n)/(2) * \left [2 * a + (n - 1)* d \right ]`

Substituting the values of a and d gives;

`S_n = (n)/(2) * \left [2 * 138 + (n - 1)* 55 \right ] = (n)/(2) * \left [221 + n * 55 \right ] = 110.5\cdot n + 27.5 \cdot n^2`

∴ 3) The amount of money she has after n weeks = Sₙ = 27.5·n² + 110.5·n.