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

This problem describes an arithmetic sequence, where Vanshika planted a certain number of seeds each month, increasing by a fixed amount each time.

### Step 1: Recognizing the arithmetic sequence
- **First term (a₁)**: In the first month, she planted 5 seeds. So, \( a₁ = 5 \).
- **Last term (aₙ)**: In the final month, she planted 50 seeds. So, \( aₙ = 50 \).
- **Common difference (d)**: Each month she planted 3 more seeds than the previous month. So, \( d = 3 \).

### Step 2: Using the formula for the nth term
The formula for the nth term of an arithmetic sequence is:
\[
aₙ = a₁ + (n - 1)d
\]
We know \( aₙ = 50 \), \( a₁ = 5 \), and \( d = 3 \). Now, we can solve for \( n \), the number of months:

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

Simplifying:

\[
50 = 5 + 3n - 3
\]
\[
50 = 3n + 2
\]
\[
48 = 3n
\]
\[
n = 16
\]

So, she planted seeds for **16 months**.

### Step 3: Finding the total number of seeds
The sum of an arithmetic sequence is given by the formula:
\[
Sₙ = \frac{n}{2} \times (a₁ + aₙ)
\]
Substituting the values \( n = 16 \), \( a₁ = 5 \), and \( aₙ = 50 \):

\[
S₁₆ = \frac{16}{2} \times (5 + 50)
\]
\[
S₁₆ = 8 \times 55 = 440
\]

### Conclusion:
Vanshika planted a total of **440 seeds**.

Would you like further details or clarification on any steps?

Here are 5 related questions to explore:
1. What if Vanshika planted 4 more seeds every month instead of 3?
2. How would the total change if she planted 10 seeds in the first month instead of 5?
3. How many months would it take for Vanshika to plant 100 seeds in the final month?
4. Can you generalize a formula for the total seeds if the common difference and initial seeds vary?
5. How would the sum change if she decreased the number of seeds each month instead?

**Tip:** In an arithmetic sequence, the sum can be found quickly by multiplying the average of the first and last terms by the number of terms.

Vanshika decided to plant a certain number of seeds every month as a part of the gardening project. In the first month she planted 5 seeds and in 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?

Discover how to calculate the total number of seeds planted by Vanshika using the arithmetic sequence formula. The problem involves increasing the number of seeds by 3 each month, starting from 5 and ending at 50 seeds. Suitable for students in Grades 6-8.

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