https://file.aiquickdraw.com/mathbot/file/17261344749502wwtnvzy.jpeg

The problem you provided involves an arithmetic progression (AP), where the gifts given by the husband increase every year. Let's break it down:

### Given:
- On their wedding day (year 0), he gave Php 400.
- On the 1st wedding anniversary, he gave Php 800.
- On the 2nd wedding anniversary, he gave Php 1600, and so on.

This shows that the gift doubles every year, forming a geometric progression (GP) where the first term \( a = 400 \) and the common ratio \( r = 2 \).

#### Part A: How much would she receive on their 9th wedding anniversary?

For a geometric sequence, the formula to calculate the \( n \)-th term is:
\[
T_n = a \cdot r^{n-1}
\]
Here, \( a = 400 \), \( r = 2 \), and for the 9th wedding anniversary, \( n = 9 \).

\[
T_9 = 400 \cdot 2^{9-1} = 400 \cdot 2^8
\]

#### Part B: Compute the total amount the wife had received as gifts from their wedding day up to their 9th wedding anniversary.

The sum of the first \( n \) terms in a geometric progression is given by the formula:
\[
S_n = a \cdot \frac{1 - r^n}{1 - r}
\]
Here, \( a = 400 \), \( r = 2 \), and \( n = 10 \) (since you start counting from year 0 for the wedding day).

Let me calculate both A and B for you.### Solutions:
- **Part A**: On their 9th wedding anniversary, the wife would receive **Php 102,400**.
- **Part B**: The total amount she would have received from their wedding day up to the 9th wedding anniversary is **Php 409,200**.

Would you like more details on how these were calculated, or do you have any further questions?

Here are 5 related questions to consider:
1. What would be the total amount received by the 10th wedding anniversary?
2. How would the formula change if the gift increased by a fixed amount instead of doubling?
3. Can you derive the formula for the \(n\)-th term of an arithmetic progression (AP) if the increase was fixed each year?
4. How does the geometric progression formula apply to other real-life financial scenarios, like investments?
5. What happens if the common ratio changes, for example, if the gift triples instead of doubling?

**Tip**: Geometric progressions are useful in calculating compound interest and exponential growth scenarios!

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

Explore how to calculate the gift amounts a wife receives using geometric progression. This example shows the growth in gifts over 9 years, with the total amount calculated using geometric progression formulas. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation