Arithmetic Progression
Arithmetic Progression
Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive numbers is constant. This constant difference is called the common difference (d).
The general formula for the nth term of an AP is:
$T_n = a + (n-1)d$
where a is the first term, n is the number of the term, and d is the common difference.
For example, if the first term (a) is 5 and the common difference (d) is 3, then the 10th term $(T_{10})$ of the AP is:
$T_{10} = 5 + (10-1)3 = 5 + 9 \times 3 = 32$
The sum of the first n terms of an AP is given by the formula: $S_n = \dfrac{n}{2}(2a + (n-1)d)$ where a is the first term, n is the number of terms, and d is the common difference.
AP’s have various applications in mathematics and real-life scenarios, such as calculating the sum of a series of numbers, modeling linear growth, and solving problems involving consecutive integers.
What is Arithmetic Progression?
Arithmetic Progression (AP)
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive numbers is the same. This common difference is often denoted by ’d'.
The general formula for the nth term of an AP is:
$ T_n = a + (n-1)d $
where:
- $T_n$ is the nth term of the AP
- a is the first term of the AP
- d is the common difference
- n is the number of the term
Example:
Consider the following sequence of numbers:
2, 5, 8, 11, 14, …
This sequence is an AP with a common difference of 3. The first term is 2, and the second term is 5. The difference between these two terms is 3. The third term is 8, and the difference between the second and third terms is also 3. This pattern continues for all the terms in the sequence.
Properties of AP:
- The sum of the first n terms of an AP is given by the formula:
$ S_n = \dfrac{n}{2}(a + T_n) $
where:
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$S_n$ is the sum of the first n terms of the AP
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a is the first term of the AP
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$T_n$ is the nth term of the AP
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n is the number of terms
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The arithmetic mean of an AP is given by the formula:
$ A = \dfrac{(a + T_n)}{2} $
where:
- A is the arithmetic mean of the AP
- a is the first term of the AP
- $T_n$ is the nth term of the AP
Applications of AP:
AP’s have a wide range of applications in various fields, including mathematics, physics, engineering, and economics. Some examples of applications of AP’s include:
- In mathematics, AP’s are used to study sequences and series.
- In physics, AP’s are used to study motion with constant acceleration.
- In engineering, AP’s are used to design and analyze structures.
- In economics, AP’s are used to study population growth and economic growth.
Notation in Arithmetic Progression
Notation in Arithmetic Progression
In arithmetic progression (AP), we use specific notation to represent the terms and other important elements of the progression. Here’s an explanation of the commonly used notations:
1. First Term (a):
- The first term of an AP is denoted by ‘a’.
- It represents the initial value from which the progression starts.
2. Common Difference (d):
- The common difference of an AP is denoted by ’d'.
- It represents the constant value added to each term to obtain the next term.
3. nth Term ($a_n$):
- The nth term of an AP is denoted by ’ $ a_n $ ‘.
- It represents the value of the term at the nth position in the progression.
4. General Term Formula:
- The general term formula for an AP is given by: $a_n $ = a + (n - 1)d
- This formula helps us find the value of any term in the progression based on its position (n).
5. Sum of n Terms ($S_n$):
- The sum of the first n terms of an AP is denoted by ‘$S_n$’.
- It represents the total value obtained by adding all the terms from the first term (a) to the nth term ($a_n$).
6. Sum of n Terms Formula:
- The sum of n terms formula for an AP is given by: $S_n = \dfrac{n}{2} \times (a + a_n)$
- This formula helps us find the total sum of the first n terms in the progression.
Examples:
1. Arithmetic Progression:
- Consider an AP with a = 5 and d = 3.
- The first few terms of this AP are: 5, 8, 11, 14, 17, …
2. Finding the nth Term:
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To find the $10^{th}$ term ($a_{10}$) of the AP with a = 5 and d = 3, we use the formula:
$a_n = a + (n - 1)d$
$\Rightarrow a_{10} = 5 + (10 - 1)3$
$\Rightarrow a_{10} = 5 + 9 \times 3$
$\Rightarrow a_{10} = 5 + 27$
$\Rightarrow a_{10} = 32$
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Therefore, the 10th term of the AP is 32.
3. Finding the Sum of n Terms:
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To find the sum of the first 10 terms ($S_{10}$) of the AP with a = 5 and d = 3, we use the formula:
$S_n $= $\dfrac{n}{2} \times (a + a_n)$
$\Rightarrow S_{10}$ = $\dfrac{10}{2} \times (5 + 32)$
$\Rightarrow S_{10} = 5 \times 37$
$\Rightarrow S_{10} = 185$
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Therefore, the sum of the first 10 terms of the AP is 185.
In summary, the notation used in arithmetic progression helps us represent and manipulate the terms, common difference, nth term, and sum of terms in a clear and concise manner.
General Form of an AP
General Form of an Arithmetic Progression (AP)
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is the same. This constant difference is called the common difference of the AP.
The general form of an AP is:
a, a + d, a + 2d, a + 3d, …, a + nd
where:
- a is the first term of the AP.
- d is the common difference of the AP.
- n is the number of terms in the AP.
For example, consider the AP 3, 7, 11, 15, 19, …. Here, the first term (a) is 3, and the common difference (d) is 4. So, the general form of this AP is:
3, 3 + 4, 3 + 2(4), 3 + 3(4), 3 + 4(4), …
Simplifying this expression, we get:
3, 7, 11, 15, 19, …
which is the given AP.
Properties of an AP
- The sum of the first n terms of an AP is given by the formula:
$S_n = \dfrac{n}{2}(2a + (n - 1)d)$
where:
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$S_n$ is the sum of the first n terms of the AP.
-
a is the first term of the AP.
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d is the common difference of the AP.
-
n is the number of terms in the AP.
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The nth term of an AP is given by the formula:
$T_n = a + (n - 1)d$
where:
- $T_n$ is the nth term of the AP.
- a is the first term of the AP.
- d is the common difference of the AP.
- n is the number of the term in the AP.
Examples of AP’s
- The sequence 1, 3, 5, 7, 9, … is an AP with a = 1 and d = 2.
- The sequence 2, 4, 6, 8, 10, … is an AP with a = 2 and d = 2.
- The sequence 3, 6, 9, 12, 15, … is an AP with a = 3 and d = 3.
Applications of AP’s
AP’s are used in a variety of applications, including:
- Finding the sum of a series of numbers.
- Calculating the average of a set of numbers.
- Modeling real-world phenomena, such as population growth and radioactive decay.
Sum of N Terms of AP
The sum of N terms of an arithmetic progression (AP) is a fundamental concept in mathematics, particularly in the study of sequences and series. An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference (d).
The formula for the sum of N terms of an AP is given by:
$ S_N = \dfrac{N}{2}\times (2a + (N - 1)\times d) $
where:
- $S_N $ represents the sum of the first N terms of the AP.
- a is the first term of the AP.
- d is the common difference of the AP.
- N is the number of terms to be summed.
To understand this formula better, let’s consider an example. Suppose we have an AP with a first term (a) of 5 and a common difference (d) of 3. We want to find the sum of the first 10 terms (N = 10) of this AP.
Plugging these values into the formula, we get:
$ S_{10} = \dfrac{10}{2} \times (2 \times 5 + (10 - 1) \times 3) $
Simplifying the expression:
$ S_{10} = 5 \times (10 + 9 \times 3) $
$ S_{10} = 5 \times (10 + 27) $
$ S_{10} = 5 \times 37 $
$ S_{10} = 185 $
Therefore, the sum of the first 10 terms of the AP with a first term of 5 and a common difference of 3 is 185.
This formula can be used to find the sum of N terms of any AP, given the first term (a) and the common difference (d). It is a useful tool in various mathematical applications, including calculating the total distance traveled by an object moving with constant acceleration, finding the sum of a series of payments or deposits, and many more.
Arithmetic Progressions Solved Examples
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive numbers is the same. The first term of an AP is called the first term, and the common difference between any two consecutive terms is called the common difference.
Example 1:
Find the first four terms of an AP with first term 5 and common difference 3.
Solution:
The first term of the AP is 5. The second term is 5 + 3 = 8. The third term is 8 + 3 = 11. The fourth term is 11 + 3 = 14.
Therefore, the first four terms of the AP are 5, 8, 11, and 14.
Example 2:
Find the sum of the first 10 terms of an AP with first term 7 and common difference 2.
Solution:
The formula for the sum of the first n terms of an AP is:
$$S_n = \frac{n}{2}(2a + (n - 1)d)$$
where:
- Sn is the sum of the first n terms
- a is the first term
- d is the common difference
- n is the number of terms
In this case, a = 7, d = 2, and n = 10. Substituting these values into the formula, we get:
$$S_{10} = \frac{10}{2}(2(7) + (10 - 1)2)$$ $$S_{10} = 5(14 + 9(2))$$ $$S_{10} = 5(32)$$ $$S_{10} = 160$$
Therefore, the sum of the first 10 terms of the AP is 160.
Example 3:
Find the 15th term of an AP with first term 4 and common difference 5.
Solution:
The formula for the nth term of an AP is:
$$T_n = a + (n - 1)d$$
where:
- Tn is the nth term
- a is the first term
- d is the common difference
- n is the number of the term
In this case, a = 4, d = 5, and n = 15. Substituting these values into the formula, we get:
$$T_{15} = 4 + (15 - 1)5$$ $$T_{15} = 4 + 14(5)$$ $$T_{15} = 4 + 70$$ $$T_{15} = 74$$
Therefore, the 15th term of the AP is 74.
Frequently Asked Questions on Arithmetic Progression
What is the general form of Arithmetic Progression?
General Form of Arithmetic Progression (AP)
In an arithmetic progression (AP), the difference between any two consecutive terms is constant. This constant difference is known as the common difference (d).
The general form of an AP is:
a, a + d, a + 2d, a + 3d, …, a + nd
where:
- a is the first term of the AP
- d is the common difference
- n is the number of terms in the AP
Examples:
- The sequence 1, 3, 5, 7, 9 is an AP with a = 1 and d = 2.
- The sequence 10, 7, 4, 1, -2 is an AP with a = 10 and d = -3.
- The sequence 0.5, 1, 1.5, 2, 2.5 is an AP with a = 0.5 and d = 0.5.
Properties of AP:
- The sum of the first n terms of an AP is given by the formula:
$ S_n = \dfrac{n}{2}(2a + (n - 1)d) $
- The nth term of an AP is given by the formula:
$ T_n = a + (n - 1)d $
- The common difference of an AP can be found by subtracting the first term from the second term, or by subtracting any two consecutive terms.
Applications of AP:
- AP’s are used in a variety of applications, including:
- Finding the sum of a series of numbers
- Calculating the average of a set of numbers
- Predicting future values based on past trends
- Solving problems involving interest rates and annuities
What is arithmetic progression? Give an example.
Arithmetic Progression
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive numbers is the same. This constant difference is called the common difference of the arithmetic progression.
The general formula for the nth term of an arithmetic progression is:
$$T_n = a + (n-1)d$$
where:
- $T_n$ is the nth term of the arithmetic progression
- $a$ is the first term of the arithmetic progression
- $d$ is the common difference of the arithmetic progression
- $n$ is the number of the term
Example:
Consider the following sequence of numbers:
$$2, 5, 8, 11, 14, 17, 20$$
This sequence is an arithmetic progression with a common difference of 3. The first term of the sequence is 2, and the second term is 5. The difference between these two terms is 3. The third term is 8, and the difference between the second and third terms is also 3. This pattern continues for the rest of the sequence.
Applications of Arithmetic Progressions
Arithmetic progressions have a variety of applications in mathematics and other fields. Some examples include:
- In physics, arithmetic progressions are used to model the motion of objects in uniform acceleration.
- In finance, arithmetic progressions are used to calculate the future value of an investment.
- In statistics, arithmetic progressions are used to calculate the mean and standard deviation of a data set.
Arithmetic progressions are a fundamental concept in mathematics with a wide range of applications. By understanding the concept of arithmetic progressions, you can gain a deeper understanding of many different areas of mathematics and other fields.
How to find the sum of arithmetic progression?
How to Find the Sum of an Arithmetic Progression (AP)
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive numbers is the same. For example, the sequence 1, 3, 5, 7, 9 is an AP with a common difference of 2.
The sum of an AP is the sum of all the numbers in the sequence. For example, the sum of the AP 1, 3, 5, 7, 9 is 1 + 3 + 5 + 7 + 9 = 25.
There are two formulas that can be used to find the sum of an AP:
- The sum of n terms of an AP is given by the formula:
$ S_n = \dfrac{n}{2}(2a + (n - 1)d) $
where:
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Sn is the sum of the first n terms of the AP
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a is the first term of the AP
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d is the common difference of the AP
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n is the number of terms in the AP
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The sum of an infinite AP is given by the formula:
$ S = \dfrac{a}{(1 - r)} $
where:
- S is the sum of the infinite AP
- a is the first term of the AP
- r is the common ratio of the AP
Examples:
- Find the sum of the first 10 terms of the AP 1, 3, 5, 7, 9.
$ \begin{aligned} &S_n = \dfrac{n}{2}(2a + (n - 1)d)\ &S_n = \dfrac{10}{2}(2(1) + (10 - 1)(2))\ &S_n = 5(2 + 9(2))\ &S_n = 5(20)\ &S_n = 100 \end{aligned} $
Therefore, the sum of the first 10 terms of the AP 1, 3, 5, 7, 9 is 100.
- Find the sum of the infinite AP $1, \dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{8}, ….$
$\begin{aligned} & S = \dfrac{a}{(1 - r)}\ & S = \dfrac{1}{(1 - 1/2)}\ & S = \dfrac{1}{(1/2)}\ & S = 2 \end{aligned} $
Therefore, the sum of the infinite AP $1, \dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{8}, ….$ is 2.
What are the types of progressions in Maths?
Progressions are sequences of numbers in which each term after the first is obtained by adding or subtracting a fixed number, called the common difference. There are three main types of progressions: arithmetic, geometric, and harmonic.
1. Arithmetic Progression (AP)
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is the same. The common difference can be positive or negative.
For example, the sequence 1, 3, 5, 7, 9 is an arithmetic progression with a common difference of 2.
The general formula for the nth term of an arithmetic progression is:
$ T_n = a + (n - 1)d $
where:
- $T_n$ is the nth term of the progression
- a is the first term of the progression
- d is the common difference
2. Geometric Progression (GP)
A geometric progression is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed number, called the common ratio. The common ratio can be positive or negative.
For example, the sequence 1, 2, 4, 8, 16 is a geometric progression with a common ratio of 2.
The general formula for the nth term of a geometric progression is:
$ T_n = ar^{(n - 1)} $
where:
- $T_n$ is the nth term of the progression
- a is the first term of the progression
- r is the common ratio
3. Harmonic Progression (HP)
A harmonic progression is a sequence of numbers in which the reciprocals of the terms form an arithmetic progression.
For example, the sequence $1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}$ is a harmonic progression.
The general formula for the nth term of a harmonic progression is:
$ T_n = \dfrac{1}{(a + (n - 1)d)} $
where:
- $T_n$ is the nth term of the progression
- a is the first term of the progression
- d is the common difference of the reciprocals of the terms
Examples of Progressions in Real Life
Progressions are used in a variety of real-life applications, including:
- Finance: The interest on a loan or investment can be calculated using an arithmetic progression.
- Physics: The distance traveled by an object in motion can be calculated using a geometric progression.
- Music: The notes in a musical scale form a harmonic progression.
Progressions are a powerful tool for modeling and understanding a variety of phenomena in the real world.
What is the use of Arithmetic Progression?
Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive numbers is the same. This constant difference is known as the common difference (d). The general formula for the nth term of an AP is:
nth term = first term + (n - 1) * common difference
For example, consider the AP 2, 5, 8, 11, 14, …. Here, the first term (a) is 2, and the common difference (d) is 3. The 10th term of this AP can be calculated as:
$10^{th}\ \text{term} = 2 + (10 - 1) \times 3$
$10^{th}\ \text{term} = 2 + 9 \times 3$
$10^{th}\ \text{term} = 2 + 27$
$10^{th}\ \text{term} = 29$
AP’s have various applications in different fields:
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Mathematics: AP’s are used to study sequences and series, as well as to solve problems involving summation of terms.
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Physics: AP’s are used to describe motion with constant acceleration. For example, if an object is moving with an initial velocity (u) and constant acceleration (a), then the distance (s) traveled by the object after time (t) is given by the formula:
$s = u \times t + 0.5 \times a \times t^2$
This formula is derived using AP.
- Finance: AP’s are used to calculate compound interest. In compound interest, the interest earned in each year is added to the principal amount, and interest is calculated on the increased amount in subsequent years. The formula for compound interest is:
$A = P \times (1 + \dfrac{r}{n})^{(n\times t)}$
where:
A = final amount
P = principal amount
r = annual interest rate
n = number of times interest is compounded per year
t = number of years
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Statistics: AP’s are used to calculate measures of central tendency, such as mean, median, and mode.
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Engineering: AP’s are used in various engineering applications, such as calculating the forces acting on structures and designing gears and pulleys.
In summary, Arithmetic Progression is a useful concept with applications in various fields, including mathematics, physics, finance, statistics, and engineering.