Ridhima Saved₹ 2 In The First Week Of June (2019), ₹4 In The Second Week, ₹8 In The Third Week Of The Month And So On. What Will Be Her Savings At The End Of The Second
Ridhima's Savings: A Geometric Progression Problem
In this article, we will explore the concept of geometric progression and how it can be applied to real-life problems. We will use the example of Ridhima's savings to demonstrate how to calculate the total savings at the end of a given period.
What is Geometric Progression?
Geometric progression is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In other words, it is a sequence of numbers in which each term is obtained by multiplying the previous term by a constant factor.
Ridhima's Savings
Ridhima saved ₹2 in the first week of June 2019, ₹4 in the second week, ₹8 in the third week, and so on. This is an example of a geometric progression with a first term of ₹2 and a common ratio of 2.
Calculating the Total Savings
To calculate the total savings at the end of the second month, we need to find the sum of the savings for all four weeks in June and all four weeks in July.
Weeks in June
The savings for the four weeks in June are:
- Week 1: ₹2
- Week 2: ₹4
- Week 3: ₹8
- Week 4: ₹16
The total savings for June is the sum of these four weeks:
₹2 + ₹4 + ₹8 + ₹16 = ₹30
Weeks in July
The savings for the four weeks in July are:
- Week 1: ₹32
- Week 2: ₹64
- Week 3: ₹128
- Week 4: ₹256
The total savings for July is the sum of these four weeks:
₹32 + ₹64 + ₹128 + ₹256 = ₹480
Total Savings
The total savings at the end of the second month is the sum of the total savings for June and July:
₹30 + ₹480 = ₹510
In this article, we used the example of Ridhima's savings to demonstrate how to calculate the total savings at the end of a given period using geometric progression. We found that the total savings at the end of the second month is ₹510.
Geometric Progression Formula
The formula for the sum of a geometric progression is:
S = a(1 - r^n) / (1 - r)
where:
- S is the sum of the geometric progression
- a is the first term
- r is the common ratio
- n is the number of terms
Example Problem
Find the sum of the geometric progression 2, 4, 8, 16, ...
Using the formula, we get:
S = 2(1 - 2^4) / (1 - 2) = 2(1 - 16) / (-1) = 2(-15) / (-1) = 30
Therefore, the sum of the geometric progression is 30.
Real-Life Applications
Geometric progression has many real-life applications, including:
- Compound interest: When interest is compounded annually, the interest earned in each year is a geometric progression.
- Population growth: The population of a country can be modeled using a geometric progression.
- Investment: The value of an investment can be modeled using a geometric progression.
In conclusion, geometric progression is a powerful tool for modeling real-life problems. It has many applications in finance, population growth, and investment. We used the example of Ridhima's savings to demonstrate how to calculate the total savings at the end of a given period using geometric progression.
Ridhima's Savings: A Geometric Progression Problem - Q&A
In our previous article, we explored the concept of geometric progression and how it can be applied to real-life problems using the example of Ridhima's savings. In this article, we will answer some frequently asked questions related to geometric progression and Ridhima's savings.
Q: What is the formula for the sum of a geometric progression?
A: The formula for the sum of a geometric progression is:
S = a(1 - r^n) / (1 - r)
where:
- S is the sum of the geometric progression
- a is the first term
- r is the common ratio
- n is the number of terms
Q: How do I calculate the sum of a geometric progression?
A: To calculate the sum of a geometric progression, you need to know the first term, the common ratio, and the number of terms. You can use the formula above to calculate the sum.
Q: What is the difference between an arithmetic progression and a geometric progression?
A: An arithmetic progression is a sequence of numbers in which each term after the first is found by adding a fixed, non-zero number called the common difference. A geometric progression is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Q: Can I use geometric progression to model real-life problems?
A: Yes, geometric progression can be used to model real-life problems such as compound interest, population growth, and investment.
Q: How do I determine the common ratio in a geometric progression?
A: The common ratio in a geometric progression can be determined by dividing any term by its previous term.
Q: Can I use geometric progression to calculate the total savings at the end of a given period?
A: Yes, geometric progression can be used to calculate the total savings at the end of a given period. We used the example of Ridhima's savings to demonstrate how to calculate the total savings at the end of the second month.
Q: What is the total savings at the end of the second month for Ridhima's savings?
A: The total savings at the end of the second month for Ridhima's savings is ₹510.
Q: Can I use geometric progression to model population growth?
A: Yes, geometric progression can be used to model population growth. The population of a country can be modeled using a geometric progression.
Q: How do I calculate the population growth using geometric progression?
A: To calculate the population growth using geometric progression, you need to know the initial population, the growth rate, and the time period. You can use the formula for the sum of a geometric progression to calculate the population at the end of the given time period.
In conclusion, geometric progression is a powerful tool for modeling real-life problems. It has many applications in finance, population growth, and investment. We used the example of Ridhima's savings to demonstrate how to calculate the total savings at the end of a given period using geometric progression. We also answered some frequently asked questions related to geometric progression and Ridhima's savings.
Real-Life Applications
Geometric progression has many real-life applications, including:
- Compound interest: When interest is compounded annually, the interest earned in each year is a geometric progression.
- Population growth: The population of a country can be modeled using a geometric progression.
- Investment: The value of an investment can be modeled using a geometric progression.
Example Problem
Find the sum of the geometric progression 2, 4, 8, 16, ...
Using the formula, we get:
S = 2(1 - 2^4) / (1 - 2) = 2(1 - 16) / (-1) = 2(-15) / (-1) = 30
Therefore, the sum of the geometric progression is 30.
In conclusion, geometric progression is a powerful tool for modeling real-life problems. It has many applications in finance, population growth, and investment. We used the example of Ridhima's savings to demonstrate how to calculate the total savings at the end of a given period using geometric progression. We also answered some frequently asked questions related to geometric progression and Ridhima's savings.