Let's solve the problem step by step.

Part (a): Writing a formula for the sequence

Charlie initially deposits $115 into the account. Each week thereafter, he deposits $35. This forms an arithmetic sequence where:

• The first term $a$ is $115$.
• The common difference $d$ is $35$.

The general formula for an arithmetic sequence is:

$a_n = a + (n - 1) \cdot d$

Where:

• $a_n$ is the amount deposited after $n$ weeks,
• $a$ is the initial amount ($115),
• $d$ is the amount added each week ($35).

So, the formula becomes:

$a_n = 115 + (n - 1) \cdot 35$

Part (b): Total money deposited after 30 weeks

To find the total deposited after 30 weeks, we need to sum the sequence up to $n = 30$.

The sum $S_n$ of the first $n$ terms of an arithmetic sequence is:

$S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d)$

Plugging in the values:

• $n = 30$
• $a = 115$
• $d = 35$

$S_{30} = \frac{30}{2} \cdot (2 \cdot 115 + (30 - 1) \cdot 35)$ $S_{30} = 15 \cdot (230 + 1015)$ $S_{30} = 15 \cdot 1245$ $S_{30} = 18675$

So, the total money deposited after 30 weeks is $18,675.

Would you like further details or have any questions?

Related Questions:

1. What is the total amount deposited after 10 weeks?
2. What if the initial deposit was $200 instead of $115? How would the formula change?
3. How much would be deposited after 52 weeks (1 year)?
4. How would the sum formula look if the deposit amount per week increased by $5 each month?
5. Can you write the recursive formula for this sequence?

Tip:

Always identify whether a problem involves arithmetic or geometric sequences by examining whether the change between terms is constant (arithmetic) or multiplicative (geometric).