This problem follows an arithmetic sequence where Vanshika plants seeds each month, with the number of seeds increasing by 3 each month.

The first month, Vanshika plants 5 seeds, and by the final month, she plants 50 seeds. We are asked to find the total number of seeds planted.

Steps to solve:

1. The general form of an arithmetic sequence is: $a_n = a_1 + (n - 1)d$ where:

   • $a_1 = 5$ (the number of seeds planted in the first month),
   • $a_n = 50$ (the number of seeds planted in the final month),
   • $d = 3$ (the common difference in the number of seeds planted each month).

2. Use the equation of the $n$-th term of an arithmetic sequence to find $n$ (the number of months). $a_n = a_1 + (n - 1)d$ Plugging in the known values: $50 = 5 + (n - 1) \cdot 3$ Simplifying: $50 = 5 + 3(n - 1)$ $50 - 5 = 3(n - 1)$ $45 = 3(n - 1)$ $15 = n - 1$ $n = 16$ So, there are 16 months.

3. To find the total number of seeds planted, use the sum formula for an arithmetic sequence: $S_n = \frac{n}{2}(a_1 + a_n)$ where $S_n$ is the total number of seeds planted, $n = 16$, $a_1 = 5$, and $a_n = 50$.

   Plugging in the values: $S_{16} = \frac{16}{2}(5 + 50)$ $S_{16} = 8 \times 55 = 440$

Conclusion:

Vanshika planted a total of 440 seeds.

Would you like further clarification or details?

Here are 5 related questions:

1. What is an arithmetic sequence?
2. How do you find the sum of an arithmetic sequence?
3. Can you derive the general term formula for any arithmetic sequence?
4. What would happen if the common difference in seed planting changed?
5. How would you calculate the sum if the number of months was unknown?

Tip: Arithmetic sequences have a constant difference between consecutive terms, which makes calculating totals simple using the sum formula!