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

This problem involves an arithmetic sequence because Vanshika plants more seeds every month, with the same difference of 3 seeds between consecutive months.

### Given:
- First month seeds (\(a_1\)) = 5
- Last month seeds (\(a_n\)) = 50
- Common difference (\(d\)) = 3

To find:
1. **Number of months** she planted seeds.
2. **Total number of seeds** planted.

#### 1. Finding the number of months (n):
The formula for the \(n\)-th term of an arithmetic sequence is:
\[
a_n = a_1 + (n-1) \cdot d
\]
Substitute the known values:
\[
50 = 5 + (n-1) \cdot 3
\]
Simplify:
\[
50 = 5 + 3n - 3
\]
\[
50 = 3n + 2
\]
\[
48 = 3n
\]
\[
n = \frac{48}{3} = 16
\]
So, Vanshika planted seeds for **16 months**.

#### 2. Total number of seeds planted:
The sum of the first \(n\) terms of an arithmetic sequence is given by:
\[
S_n = \frac{n}{2} \cdot (a_1 + a_n)
\]
Substitute the known values:
\[
S_{16} = \frac{16}{2} \cdot (5 + 50)
\]
\[
S_{16} = 8 \cdot 55 = 440
\]
So, the total number of seeds Vanshika planted is **440 seeds**.

Would you like further details or have any questions?

Here are some related questions to expand on this topic:
1. What if the difference between seeds planted each month was 4 instead of 3?
2. How would the total change if Vanshika planted 8 seeds in the first month?
3. How would you calculate the total if the number of seeds planted each month doubled instead of increasing by a constant number?
4. What would happen if she planted fewer seeds each month instead of more?
5. Can we derive a formula to calculate the total seeds for any starting number and difference?

**Tip**: In arithmetic sequences, the common difference can significantly change both the final term and the sum of terms.

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.

Solve an arithmetic sequence problem where Vanshika plants seeds each month, starting with 5 seeds in the first month and 50 seeds in the final month. Learn how to find the number of months and total seeds planted using key formulas for arithmetic sequences. Suitable for students in Grades 6-8.

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