Find The 17th Term Of The Arithmetic Progression (A.P.): -6, -1,
Introduction to Arithmetic Progression (A.P.)
An arithmetic progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. The general form of an A.P. is given by:
a, a + d, a + 2d, a + 3d, ...
where 'a' is the first term and 'd' is the common difference.
Understanding the Given A.P.
In this problem, we are given an A.P. with the first term 'a' as -6 and the common difference 'd' as 5. The A.P. is given as:
-6, -1, 6, 13, ...
We need to find the 17th term of this A.P.
Formula for the nth Term of an A.P.
The formula for the nth term of an A.P. is given by:
an = a + (n - 1)d
where 'an' is the nth term, 'a' is the first term, 'n' is the term number, and 'd' is the common difference.
Finding the 17th Term
To find the 17th term, we can use the formula:
an = a + (n - 1)d
Substituting the values, we get:
a17 = -6 + (17 - 1)5 a17 = -6 + 16(5) a17 = -6 + 80 a17 = 74
Therefore, the 17th term of the given A.P. is 74.
Example and Practice
Let's consider another example to understand the concept better.
Find the 10th term of the A.P. 3, 6, 9, 12, ...
Using the formula, we get:
a10 = 3 + (10 - 1)3 a10 = 3 + 9(3) a10 = 3 + 27 a10 = 30
Therefore, the 10th term of the given A.P. is 30.
Real-World Applications of A.P.
Arithmetic Progression (A.P.) has numerous real-world applications in various fields such as finance, economics, and engineering.
In finance, A.P. is used to calculate the future value of an investment or a loan. For example, if you invest $1000 at a 5% interest rate, the future value after 5 years can be calculated using the A.P. formula.
In economics, A.P. is used to model the growth of a population or the demand for a product. For example, if the population of a city is growing at a rate of 2% per year, the population after 10 years can be calculated using the A.P. formula.
In engineering, A.P. is used to design and optimize systems such as bridges, buildings, and electrical circuits.
Conclusion
In conclusion, finding the 17th term of an arithmetic progression (A.P.) is a simple process that involves using the formula for the nth term of an A.P. The formula is given by:
an = a + (n - 1)d
where 'an' is the nth term, 'a' is the first term, 'n' is the term number, and 'd' is the common difference.
By understanding the concept of A.P. and using the formula, we can find the nth term of any A.P. and apply it to real-world problems.
Frequently Asked Questions (FAQs)
Q: What is an arithmetic progression (A.P.)?
A: An arithmetic progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant.
Q: What is the formula for the nth term of an A.P.?
A: The formula for the nth term of an A.P. is given by:
an = a + (n - 1)d
where 'an' is the nth term, 'a' is the first term, 'n' is the term number, and 'd' is the common difference.
Q: How do I find the 17th term of an A.P.?
A: To find the 17th term of an A.P., you can use the formula:
an = a + (n - 1)d
Substitute the values of 'a', 'n', and 'd' into the formula to find the 17th term.
Q: What are the real-world applications of A.P.?
A: Arithmetic Progression (A.P.) has numerous real-world applications in various fields such as finance, economics, and engineering.
Q: Can I use A.P. to model the growth of a population?
A: Yes, you can use A.P. to model the growth of a population. For example, if the population of a city is growing at a rate of 2% per year, the population after 10 years can be calculated using the A.P. formula.
Q: Can I use A.P. to design and optimize systems?
A: Yes, you can use A.P. to design and optimize systems such as bridges, buildings, and electrical circuits.
Q: What are some common mistakes to avoid when working with A.P.?
A: Some common mistakes to avoid when working with A.P. include:
- Not using the correct formula for the nth term of an A.P.
- Not substituting the correct values into the formula.
- Not checking the units of the answer.
By avoiding these common mistakes, you can ensure that your calculations are accurate and reliable.
Q: What is the difference between an arithmetic progression (A.P.) and a geometric progression (G.P.)?
A: An arithmetic progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant. On the other hand, a geometric progression (G.P.) is a sequence of numbers in which the ratio between any two consecutive terms is constant.
Q: How do I determine if a sequence is an arithmetic progression (A.P.) or a geometric progression (G.P.)?
A: To determine if a sequence is an arithmetic progression (A.P.) or a geometric progression (G.P.), you can check if the difference between any two consecutive terms is constant (A.P.) or if the ratio between any two consecutive terms is constant (G.P.).
Q: What is the formula for the sum of the first n terms of an arithmetic progression (A.P.)?
A: The formula for the sum of the first n terms of an arithmetic progression (A.P.) is given by:
Sn = n/2 (2a + (n - 1)d)
where 'Sn' is the sum of the first n terms, 'a' is the first term, 'n' is the number of terms, and 'd' is the common difference.
Q: How do I find the sum of the first n terms of an arithmetic progression (A.P.)?
A: To find the sum of the first n terms of an arithmetic progression (A.P.), you can use the formula:
Sn = n/2 (2a + (n - 1)d)
Substitute the values of 'a', 'n', and 'd' into the formula to find the sum.
Q: What is the formula for the sum of an infinite arithmetic progression (A.P.)?
A: The formula for the sum of an infinite arithmetic progression (A.P.) is given by:
S = a/2 (1 + 1/(1 - r))
where 'S' is the sum, 'a' is the first term, and 'r' is the common ratio.
Q: How do I find the sum of an infinite arithmetic progression (A.P.)?
A: To find the sum of an infinite arithmetic progression (A.P.), you can use the formula:
S = a/2 (1 + 1/(1 - r))
Substitute the values of 'a' and 'r' into the formula to find the sum.
Q: What is the formula for the nth term of an arithmetic progression (A.P.)?
A: The formula for the nth term of an arithmetic progression (A.P.) is given by:
an = a + (n - 1)d
where 'an' is the nth term, 'a' is the first term, 'n' is the term number, and 'd' is the common difference.
Q: How do I find the nth term of an arithmetic progression (A.P.)?
A: To find the nth term of an arithmetic progression (A.P.), you can use the formula:
an = a + (n - 1)d
Substitute the values of 'a', 'n', and 'd' into the formula to find the nth term.
Q: What are some real-world applications of arithmetic progression (A.P.)?
A: Arithmetic progression (A.P.) has numerous real-world applications in various fields such as finance, economics, and engineering.
Q: Can I use arithmetic progression (A.P.) to model the growth of a population?
A: Yes, you can use arithmetic progression (A.P.) to model the growth of a population. For example, if the population of a city is growing at a rate of 2% per year, the population after 10 years can be calculated using the A.P. formula.
Q: Can I use arithmetic progression (A.P.) to design and optimize systems?
A: Yes, you can use arithmetic progression (A.P.) to design and optimize systems such as bridges, buildings, and electrical circuits.
Q: What are some common mistakes to avoid when working with arithmetic progression (A.P.)?
A: Some common mistakes to avoid when working with arithmetic progression (A.P.) include:
- Not using the correct formula for the nth term of an A.P.
- Not substituting the correct values into the formula.
- Not checking the units of the answer.
By avoiding these common mistakes, you can ensure that your calculations are accurate and reliable.
Q: How do I determine if an arithmetic progression (A.P.) is increasing or decreasing?
A: To determine if an arithmetic progression (A.P.) is increasing or decreasing, you can check the sign of the common difference 'd'. If 'd' is positive, the A.P. is increasing. If 'd' is negative, the A.P. is decreasing.
Q: Can I use arithmetic progression (A.P.) to model the decay of a substance?
A: Yes, you can use arithmetic progression (A.P.) to model the decay of a substance. For example, if a substance is decaying at a rate of 5% per year, the amount of the substance after 10 years can be calculated using the A.P. formula.
Q: Can I use arithmetic progression (A.P.) to design and optimize financial instruments?
A: Yes, you can use arithmetic progression (A.P.) to design and optimize financial instruments such as bonds, stocks, and options.
Q: What are some common applications of arithmetic progression (A.P.) in finance?
A: Arithmetic progression (A.P.) has numerous applications in finance, including:
- Calculating the future value of an investment or a loan.
- Modeling the growth of a portfolio.
- Designing and optimizing financial instruments such as bonds, stocks, and options.
Q: Can I use arithmetic progression (A.P.) to model the growth of a company?
A: Yes, you can use arithmetic progression (A.P.) to model the growth of a company. For example, if a company is growing at a rate of 10% per year, the revenue of the company after 10 years can be calculated using the A.P. formula.
Q: Can I use arithmetic progression (A.P.) to design and optimize systems in engineering?
A: Yes, you can use arithmetic progression (A.P.) to design and optimize systems in engineering such as bridges, buildings, and electrical circuits.
Q: What are some common applications of arithmetic progression (A.P.) in engineering?
A: Arithmetic progression (A.P.) has numerous applications in engineering, including:
- Designing and optimizing bridges.
- Designing and optimizing buildings.
- Designing and optimizing electrical circuits.
Q: Can I use arithmetic progression (A.P.) to model the growth of a population in a specific region?
A: Yes, you can use arithmetic progression (A.P.) to model the growth of a population in a specific region. For example, if the population of a city is growing at a rate of 2% per year, the population after 10 years can be calculated using the A.P. formula.
Q: Can I use arithmetic progression (A.P.) to design and optimize systems in economics?
A: Yes, you can use arithmetic progression (A.P.) to design and optimize systems in economics such as modeling the growth of a economy or designing and optimizing economic policies.
Q: What are some common applications of arithmetic progression (A.P.) in economics?
A: Arithmetic progression (A.P.) has numerous applications in economics, including:
- Modeling the growth of a economy.
- Designing and optimizing economic policies.
- Calculating the future value of an investment or a loan.
Q: Can I use arithmetic progression (A.P.) to model the growth of a company in a specific industry?
A: Yes, you can use arithmetic progression (A.P.) to model the growth of a company in a specific industry. For example, if a company in the technology industry is growing at a rate of 10% per year, the revenue of the company after 10 years can be calculated using the A.P. formula.
Q: Can I use arithmetic progression (A.P.) to design and optimize systems in finance?
A: Yes, you can use arithmetic progression (A.P.) to design and optimize systems in finance such as modeling the growth of a portfolio or designing and optimizing financial instruments.
Q: What are some common applications of arithmetic progression (A.P.) in finance?
A: Arithmetic progression (A.P.) has numerous applications in finance, including:
- Calculating the future value of an investment or a loan.
- Modeling the growth of a portfolio.
- Designing and optimizing financial instruments such as bonds, stocks, and options.
Q: Can I use arithmetic progression (A.P.) to model the growth of a population in a specific country?
A: Yes, you can use arithmetic progression (A.P.) to model the growth of a population in a specific country. For example, if the population of a country is growing at a rate of 2% per year, the population after 10 years can be calculated using the A.P. formula.
Q: Can I use arithmetic progression (A.P.) to design and optimize systems in engineering?
A: Yes, you can use arithmetic progression (A.P.) to design and optimize systems in engineering such as designing and optimizing bridges, buildings, and electrical circuits.
Q: What are some common applications of arithmetic progression (A.P.) in engineering?
A: Arithmetic progression (A.P.) has numerous applications in engineering, including:
- Designing and optimizing bridges