Vanshika decided to plant a certain number of seeds every month as a part of a gardening project. In the first month, she planted 5 flower seeds, and in the final month, she planted 50 flower seeds. Every month, she planted 3 more seeds than the previous month. How many flower seeds did Vanshika plant in total?

This problem follows an arithmetic sequence, where the number of flower seeds planted each month increases by a constant number (3 seeds). Let's break it down step by step.

### Given:
- In the first month, Vanshika planted 5 seeds.
- In the final month, she planted 50 seeds.
- Every month, she planted 3 more seeds than the previous month.

We can describe this situation with the following:

- First term of the arithmetic sequence (\(a_1\)) = 5
- Last term of the sequence (\(a_n\)) = 50
- Common difference (\(d\)) = 3

#### Step 1: Find the total number of months (\(n\))

The formula for the \(n\)-th term of an arithmetic sequence is:

\[
a_n = a_1 + (n - 1) \cdot d
\]

Plugging in the known values:

\[
50 = 5 + (n - 1) \cdot 3
\]

Simplify the equation:

\[
50 - 5 = (n - 1) \cdot 3
\]

\[
45 = (n - 1) \cdot 3
\]

\[
n - 1 = \frac{45}{3} = 15
\]

\[
n = 16
\]

So, Vanshika planted seeds for 16 months.

#### Step 2: Find the total number of seeds planted

The sum of the first \(n\) terms of an arithmetic sequence is given by the formula:

\[
S_n = \frac{n}{2} \cdot (a_1 + a_n)
\]

Substituting the known values:

\[
S_{16} = \frac{16}{2} \cdot (5 + 50)
\]

\[
S_{16} = 8 \cdot 55 = 440
\]

Thus, Vanshika planted a total of **440 flower seeds**.

---

Would you like more details, or do you have any other questions?

### Related Questions:
1. What would the total number of seeds be if the common difference was 4 instead of 3?
2. How many seeds would Vanshika have planted by the 10th month?
3. What if Vanshika started with 10 seeds in the first month but still increased by 3 each month?
4. How many seeds would be planted in total if she planted for 20 months instead of 16?
5. What is the formula for the sum of an arithmetic series?

### Tip:
When solving arithmetic sequence problems, always identify the first term, common difference, and number of terms before applying the sum formula.

Discover how to calculate the total number of flower seeds Vanshika planted using arithmetic sequence principles. She started with 5 seeds in the first month, increased by 3 seeds each month, and ended with 50 seeds. Find out how many seeds she planted in total in this step-by-step solution. Suitable for students in Grades 6-8.

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