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?

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

### Problem Analysis:

The amount of money the wife receives each year follows a geometric sequence, where:

- The first term (initial gift) is \( a_1 = 400 \).
- The common ratio \( r \) is \( 2 \), as the amount doubles each year.

Thus, the sequence of the amounts given each year is:
\[
400, 800, 1600, 3200, \ldots
\]
This sequence can be described by the formula for the \( n \)-th term of a geometric sequence:
\[
a_n = a_1 \cdot r^{n-1}
\]
where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

### Part A: Amount on the 9th Wedding Anniversary
To find the amount she receives on the 9th wedding anniversary, we use the formula for the \( n \)-th term of a geometric sequence:
\[
a_9 = 400 \cdot 2^{9-1} = 400 \cdot 2^8 = 400 \cdot 256 = 102,400
\]
So, she would receive **Php 102,400** on their 9th wedding anniversary.

---

### Part B: Total Amount Received from the Wedding Day to the 9th Anniversary
The total amount she received over the 9 years can be found using the sum formula for a geometric series:
\[
S_n = a_1 \cdot \frac{r^n - 1}{r - 1}
\]
where:
- \( a_1 = 400 \) (the first term),
- \( r = 2 \) (the common ratio),
- \( n = 9 \) (the number of terms).

Substituting the values:
\[
S_9 = 400 \cdot \frac{2^9 - 1}{2 - 1} = 400 \cdot (512 - 1) = 400 \cdot 511 = 204,400
\]
Thus, the total amount she received from their wedding day up to the 9th anniversary is **Php 204,400**.

### Summary:
- On the 9th wedding anniversary, the wife would receive **Php 102,400**.
- The total amount received over the 9 years is **Php 204,400**.

---

Would you like more details or have any questions?

Here are 5 questions related to the topic:
1. What if the common ratio was 3 instead of 2?
2. How would the total amount change if the first gift was Php 500 instead of Php 400?
3. How can we adjust the formula if the gift amounts decreased instead of increased?
4. What would be the total after 10 years instead of 9?
5. How can we calculate the total if the gift amounts varied randomly?

**Tip**: Geometric sequences grow rapidly, so understanding the common ratio's effect is key to predicting large terms!

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. How much would she receive on their 9th wedding anniversary? Compute the total amount the wife had received as gifts from their wedding day up to their ninth wedding anniversary?

Solve a problem involving a geometric sequence where a wife receives doubling wedding gifts each year. Learn how to calculate the amount received on the 9th wedding anniversary and the total amount over nine years. Key formulas for geometric sequences and series are used. Suitable for Grades 9-10.

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