Ricardo started a savings account for his daughter Ruth by depositing ﻿
$
5
0
0
$500﻿ into the account for her ﻿
1
1﻿st birthday. For each successive birthday, Ricardo deposits ﻿
$
2
0
0
$200﻿ more than the amount deposited for the previous birthday. This is the only money deposited into the account. What is the total amount of money Ricardo will have deposited into the account for Ruth up to and including her ﻿
6
6﻿th birthday?

This problem involves a series of deposits that follow an arithmetic sequence. Let's break down the steps to find the total amount deposited into the account up to and including Ruth's 6th birthday.

### Step 1: Understanding the Sequence
Ricardo starts by depositing $500 for Ruth's 1st birthday. For each subsequent birthday, the amount deposited increases by $200 compared to the previous year.

The sequence of deposits looks like this:
- 1st birthday: $500
- 2nd birthday: $500 + $200 = $700
- 3rd birthday: $700 + $200 = $900
- 4th birthday: $900 + $200 = $1100
- 5th birthday: $1100 + $200 = $1300
- 6th birthday: $1300 + $200 = $1500

### Step 2: Determine the Total Amount Deposited
To find the total amount deposited up to and including her 6th birthday, sum the amounts deposited each year.

\[
\text{Total Amount} = 500 + 700 + 900 + 1100 + 1300 + 1500
\]

Let's calculate the total:

\[
500 + 700 = 1200
\]
\[
1200 + 900 = 2100
\]
\[
2100 + 1100 = 3200
\]
\[
3200 + 1300 = 4500
\]
\[
4500 + 1500 = 6000
\]

### Final Answer:
The total amount of money Ricardo will have deposited into the account for Ruth up to and including her 6th birthday is **$6000**.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. What if Ricardo increased the deposit by $300 each year instead of $200?
2. How much will Ricardo deposit in total by Ruth's 10th birthday if the same pattern continues?
3. What would the total be if the initial deposit was $1000 instead of $500?
4. Can you find a general formula for the total amount deposited up to Ruth's $ n $th birthday?
5. How much would Ricardo need to deposit on Ruth's 7th birthday if the sequence continues?

**Tip:** When dealing with arithmetic sequences, summing the sequence can often be simplified using the formula for the sum of an arithmetic series.

Discover how much money Ricardo will deposit into Ruth's savings account by her 6th birthday. This problem involves an arithmetic sequence of deposits increasing by $200 each year, starting with an initial deposit of $500. Ideal for students in Grades 6-8 learning about arithmetic sequences and sum calculations.

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