7. A young man gave his wife a gift of Php 400 on their wedding day. To please her, he gave her Php 800 on their first wedding anniversary, Php 1600 on their second c wedding anniversary, and so on.

a. How much would she receive on their 9th wedding anniversary? Php 204,800

Compute the total amount the wife had received as gifts from their wedding day up to their ninth wedding anniversary? Php 408,400

Let's clarify and calculate based on the information provided:

The series of amounts given follows a **geometric sequence** where:
- The first gift \( a_1 = 400 \) on the wedding day.
- Each subsequent gift doubles, so the common ratio \( r = 2 \).

The formula for the \( n \)-th term of a geometric sequence is:
\[
a_n = a_1 \cdot r^{n-1}
\]
and the formula for the sum of the first \( n \) terms is:
\[
S_n = a_1 \cdot \frac{r^n - 1}{r - 1}
\]

### Part A: Amount on the 9th Wedding Anniversary

We need to calculate the 9th term of the sequence:

\[
a_9 = 400 \cdot 2^{9-1} = 400 \cdot 2^8 = 400 \cdot 256 = 102,400
\]
This is the correct result based on the formula. If you expected **Php 204,800**, it would suggest doubling the amount once more, but based on the doubling pattern described, the amount on the 9th anniversary should be **Php 102,400**.

### Part B: Total Amount from the Wedding Day to the 9th Anniversary

To find the total amount, we use the sum formula for a geometric series:
\[
S_9 = 400 \cdot \frac{2^9 - 1}{2 - 1} = 400 \cdot (512 - 1) = 400 \cdot 511 = 204,400
\]
So the total amount received from the wedding day to the 9th anniversary is **Php 204,400**.

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### Addressing the discrepancy in your expected answer:
- If the amount on the 9th wedding anniversary is **Php 204,800** (as you mentioned), this would indicate a different starting amount or a different pattern. However, based on the doubling sequence described, the answer should be **Php 102,400** for the 9th anniversary and **Php 204,400** for the total.

Would you like to revisit any details or clarify further?

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Here are 5 questions for deeper understanding:
1. What if the starting gift was Php 500 instead of Php 400?
2. How does changing the common ratio affect the sequence's growth?
3. What would be the total if the gifts followed an increasing arithmetic sequence instead?
4. How would the total amount change after 15 years instead of 9?
5. What if the gifts were given quarterly rather than yearly?

**Tip**: Always verify the given sequence pattern (doubling, tripling, etc.) when calculating terms and sums to avoid discrepancies!

A young man gave his wife a gift of Php 400 on their wedding day. To please her, he gave her Php 800 on their first wedding anniversary, Php 1600 on their second wedding anniversary, and so on.

 a. How much would she receive on their 9th wedding anniversary? Php 204,800

 Compute the total amount the wife had received as gifts from their wedding day up to their ninth wedding anniversary? Php 408,400

Learn how to calculate the 9th wedding anniversary gift and the total amount received from a geometric sequence of gifts starting at Php 400. This step-by-step solution covers key concepts such as the n-th term and the sum of a geometric sequence. Suitable for Grades 9-12 students.

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