Question

1) Every day, Lucy collects some candies. If Lucy collects 10 candies on the first day of Halloween, and 2 more candies each subsequent day, how many candies will Lucy have collected after 7 days?
Solve this equation word problem showing your defined variable, equation, inverse operations to solve, and final solution with units.

Answer 1

To solve the problem, we will define our variables, set up an equation, use inverse operations to find the solution, and finally present the answer clearly.

### Step 1: Define Variables
Let \( C \) be the total number of candies Lucy collects after 7 days.

### Step 2: Establish Patterns
- On the first day, Lucy collects 10 candies.
- Each subsequent day, she collects 2 more candies than the previous day.

### Step 3: Calculate Candies Collected Each Day
- Day 1: 10 candies
- Day 2: \( 10 + 2 = 12 \) candies
- Day 3: \( 12 + 2 = 14 \) candies
- Day 4: \( 14 + 2 = 16 \) candies
- Day 5: \( 16 + 2 = 18 \) candies
- Day 6: \( 18 + 2 = 20 \) candies
- Day 7: \( 20 + 2 = 22 \) candies

### Step 4: Using a General Formula
We can also express the daily collection as an arithmetic sequence:
- The amount she collects each day can be described as:
- Day \( n \): \( 10 + 2(n - 1) \)

Where \( n \) is the day number (1 through 7).

### Step 5: Total Candies Collected
To find the total candies collected after 7 days, we can use the formula for the sum of an arithmetic sequence, where:
- The first term \( a = 10 \)
- The number of terms \( n = 7 \)
- The common difference \( d = 2 \)

The \( n \)-th term can be calculated using the formula:
\[ a_n = a + (n-1)d \]
So for the 7th day:
\[ a_7 = 10 + (7 - 1) \times 2 = 10 + 12 = 22 \]

Now, we calculate the total sum of candies collected over 7 days:
\[ C = \text{Sum} = \frac{n}{2} \times (a + a_n) \]
Substitute the known values:
\[ C = \frac{7}{2} \times (10 + 22) \]
\[ C = \frac{7}{2} \times 32 \]
\[ C = 7 \times 16 = 112 \]

### Final Solution
After 7 days, Lucy will have collected a total of **112 candies**.