- **Q1:** How To Find The Middle Term Of The Arithmetic Progression (AP) -11, -7, -3, ..., 45? - **Q2:** How Many Terms Are Present In The Arithmetic Progression (AP) 7, 16, 25, ...? - **Q3:** Is -150 A Term In The Arithmetic Progression (AP) 17, 12, 7, 2, ...? Explain. - **Q4:** How To Determine An Arithmetic Progression (AP) If The 8th Term Is 31 And The 15th Term Is 16 More Than The 11th Term?

by ADMIN 400 views

Arithmetic Progressions (APs) are a fundamental concept in mathematics, particularly in sequences and series. Understanding APs is crucial for various mathematical applications and problem-solving scenarios. This comprehensive guide aims to provide a detailed explanation of APs, accompanied by solved examples to enhance your understanding and proficiency.

What is an Arithmetic Progression?

An arithmetic progression, often abbreviated as AP, 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, often denoted by d. The general form of an AP is:

a, a + d, a + 2d, a + 3d, ...

Where:

  • a is the first term,
  • d is the common difference.

Key Components of an AP

  • First Term (a): The initial value of the sequence.

  • Common Difference (d): The constant difference between consecutive terms. It can be positive, negative, or zero.

  • nth Term (an): The term at the nth position in the sequence. The formula to find the nth term is:

    an = a + (n - 1)d

  • Number of Terms (n): The total number of terms in a finite AP.

Formulas for Arithmetic Progressions

To effectively solve problems related to APs, it's essential to understand and apply the key formulas:

  1. nth Term Formula:

    an = a + (n - 1)d

    This formula allows you to find any term in the AP if you know the first term, common difference, and position of the term.

  2. Sum of n Terms Formula:

    The sum of the first n terms of an AP, denoted as Sn, can be calculated using two main formulas:

    • Sn = n/2 [2a + (n - 1)d]
    • Sn = n/2 [a + an]

    The first formula is used when you know the first term, common difference, and number of terms. The second formula is used when you know the first term, the last term (an), and the number of terms.

Solved Examples

To illustrate the concepts and formulas, let's delve into some solved examples covering common types of AP problems.

Q1: Finding the Middle Term of an A.P.

Question: Find the middle term of the A.P.

-11, -7, -3, ..., 45.

To effectively find the middle term of the arithmetic progression (AP) -11, -7, -3, ..., 45, it's crucial to systematically apply the fundamental concepts and formulas associated with APs. The middle term in an AP represents the term that lies exactly in the center of the sequence, equidistant from both ends. To determine this, we first need to ascertain the total number of terms in the sequence. This involves using the formula for the nth term of an AP, which is given by an = a + (n - 1)d, where an is the nth term, a is the first term, n is the number of terms, and d is the common difference. In this specific AP, the first term (a) is -11, and the common difference (d) can be calculated by subtracting any term from its succeeding term; for instance, -7 - (-11) equals 4. This means that each term in the sequence increases by 4. The last term of the AP is given as 45. By substituting these values into the nth term formula, we can solve for n, which will tell us the total number of terms in the sequence.

Substituting the given values, we have 45 = -11 + (n - 1)4. Simplifying this equation allows us to find the value of n. Adding 11 to both sides gives us 56 = (n - 1)4. Next, dividing both sides by 4, we get 14 = n - 1. Adding 1 to both sides, we find that n = 15. This indicates that there are 15 terms in the arithmetic progression. Now that we know the total number of terms, we can determine the position of the middle term. For a sequence with an odd number of terms, the middle term is simply the term at the position (n + 1) / 2. In this case, with 15 terms, the middle term is at the (15 + 1) / 2 = 8th position. To find the value of this middle term, we again use the nth term formula, substituting n = 8, a = -11, and d = 4. This gives us a8 = -11 + (8 - 1)4. Simplifying this, we get a8 = -11 + 7 * 4, which equals -11 + 28, resulting in a8 = 17. Therefore, the middle term of the given arithmetic progression is 17. This methodical approach of first finding the total number of terms and then identifying the middle term’s position using the formula (n + 1) / 2, followed by calculating the value of the middle term using the nth term formula, provides a clear and accurate solution to the problem.

Solution:

  1. Identify the first term (a), common difference (d), and the last term.

    • a = -11
    • d = -7 - (-11) = 4
    • Last term = 45

  2. Use the nth term formula to find the number of terms (n):

    • an = a + (n - 1)d
    • 45 = -11 + (n - 1)4
    • 56 = (n - 1)4
    • 14 = n - 1
    • n = 15

  3. Find the middle term position:

    • Middle term position = (n + 1) / 2 = (15 + 1) / 2 = 8

  4. Find the 8th term:

    • a8 = a + (8 - 1)d
    • a8 = -11 + 7 * 4
    • a8 = -11 + 28
    • a8 = 17

Answer: The middle term of the A.P. is 17.

Q2: Determining the Number of Terms in an A.P.

Question: How many terms are there in the A.P.

7, 16, 25, ... ?

To find out the number of terms in the arithmetic progression (AP) 7, 16, 25, ..., we need to first establish the pattern and identify the key parameters of the sequence. An arithmetic progression is characterized by a constant difference between consecutive terms, known as the common difference. In this AP, the first term (a) is 7. To find the common difference (d), we can subtract any term from its subsequent term. For example, subtracting 7 from 16 gives us 9, and subtracting 16 from 25 also gives us 9. Therefore, the common difference (d) in this AP is 9. This indicates that each term in the sequence increases by 9. The problem does not provide a last term, which implies that the series is being examined up to a certain unspecified point. To determine the number of terms (n) up to a specific value, we would need a defined last term. However, without this information, we can still analyze and express a general approach to solving such problems if a last term were provided.

Assuming we had a last term, we would use the formula for the nth term of an AP, which is an = a + (n - 1)d. Here, an represents the nth term (or the last term, if known), a is the first term, n is the number of terms, and d is the common difference. By substituting the values of an, a, and d into this formula, we could solve for n, thereby finding the number of terms in the AP. However, since no last term is specified in the question, we cannot compute a numerical value for n. Instead, we can illustrate the method that would be used if a last term was given. For instance, if the last term were given as, say, 100, we would substitute 100 for an, 7 for a, and 9 for d in the formula, resulting in the equation 100 = 7 + (n - 1)9. Solving this equation would give us the number of terms up to 100. In the absence of a defined last term, we can only conclude that the AP continues indefinitely, and thus, there is an infinite number of terms. However, the method explained provides a clear framework for solving similar problems when a last term is specified.

Solution:

  1. Identify the first term (a) and common difference (d):

    • a = 7
    • d = 16 - 7 = 9

  2. Without a specified last term, we cannot determine a finite number of terms.

Answer: Without a last term specified, there are infinitely many terms in the A.P.

Q3: Determining if a Number is a Term of an A.P.

Question: Determine if -150 is a term of the A.P.

17, 12, 7, 2, ...

To determine whether -150 is a term of the arithmetic progression (AP) 17, 12, 7, 2, ..., we need to apply the fundamental properties of APs and use the formula for the nth term. The key idea here is that if -150 is a term in this AP, then there must be a positive integer n such that -150 can be expressed as the nth term of the sequence. This means we need to find an n that satisfies the equation 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. First, we identify the first term (a) and the common difference (d) of the given AP. The first term (a) is 17. The common difference (d) can be found by subtracting any term from its preceding term. For instance, 12 - 17 = -5, or 7 - 12 = -5. Thus, the common difference (d) is -5. This indicates that the terms in the sequence are decreasing by 5 each time.

Now, we set an to -150 and substitute the values of a and d into the nth term formula. This gives us -150 = 17 + (n - 1)(-5). Our goal is to solve this equation for n. First, we subtract 17 from both sides of the equation, which gives us -167 = (n - 1)(-5). Next, we divide both sides by -5, resulting in 167/5 = n - 1. Calculating 167/5 gives us 33.4. Thus, the equation becomes 33.4 = n - 1. Adding 1 to both sides, we get n = 34.4. Since n represents the position of the term in the sequence, it must be a positive integer. However, our calculation yields n = 34.4, which is not an integer. This indicates that -150 cannot be a term in the given AP because there is no integer value of n that satisfies the condition. The presence of a non-integer value for n implies that -150 would fall between two terms in the sequence, which is not possible in an arithmetic progression where terms must occur at integer positions. Therefore, we conclude that -150 is not a term of the AP 17, 12, 7, 2, ...

Solution:

  1. Identify the first term (a) and common difference (d):

    • a = 17
    • d = 12 - 17 = -5

  2. Use the nth term formula:

    • an = a + (n - 1)d
    • -150 = 17 + (n - 1)(-5)

  3. Solve for n:

    • -150 = 17 - 5n + 5
    • -150 = 22 - 5n
    • -172 = -5n
    • n = 172 / 5
    • n = 34.4

  4. Since n is not a positive integer, -150 is not a term of the A.P.

Answer: -150 is not a term of the A.P.

Q4: Finding a Term Given a Condition

Question: If the 8th term of an A.P. is 31 and the 15th term is 16 more than the 11th term, find the A.P.

To find the arithmetic progression (AP), we need to determine the first term (a) and the common difference (d). The problem provides us with two crucial pieces of information: the 8th term of the AP is 31, and the 15th term is 16 more than the 11th term. We can translate these statements into mathematical equations using the formula for the nth term of an AP, which is an = a + (n - 1)d. The first statement tells us that the 8th term (a8) is 31. Substituting n = 8 into the formula, we get a8 = a + (8 - 1)d, which simplifies to 31 = a + 7d. This gives us our first equation relating a and d.

The second statement provides a relationship between the 15th term and the 11th term. It states that the 15th term is 16 more than the 11th term. We can write this as a15 = a11 + 16. Now, we express a15 and a11 in terms of a and d using the nth term formula. For the 15th term (a15), we have a15 = a + (15 - 1)d, which simplifies to a15 = a + 14d. Similarly, for the 11th term (a11), we have a11 = a + (11 - 1)d, which simplifies to a11 = a + 10d. Substituting these expressions into our equation a15 = a11 + 16, we get a + 14d = (a + 10d) + 16. Simplifying this equation, we can subtract a and 10d from both sides, resulting in 4d = 16. Dividing both sides by 4, we find that d = 4. This gives us the common difference of the AP.

Now that we have the value of d, we can substitute it back into our first equation, 31 = a + 7d, to find the value of a. Substituting d = 4, we get 31 = a + 7(4), which simplifies to 31 = a + 28. Subtracting 28 from both sides, we find that a = 3. This gives us the first term of the AP. With the first term a = 3 and the common difference d = 4, we can now write out the AP. The AP starts with 3, and each subsequent term is obtained by adding the common difference 4. Thus, the AP is 3, 7, 11, 15, ... This methodical approach of translating the given information into equations using the nth term formula, solving for the common difference, and then finding the first term, allows us to completely define the arithmetic progression.

Solution:

  1. Use the nth term formula to express the given information as equations:

    • a8 = a + 7d = 31
    • a15 = a11 + 16
    • a + 14d = a + 10d + 16

  2. Solve for d:

    • 4d = 16
    • d = 4

  3. Substitute d into the first equation to solve for a:

    • 31 = a + 7(4)
    • 31 = a + 28
    • a = 3

  4. Write the A.P. using a and d:

    • A.P.: 3, 7, 11, 15, ...

Answer: The A.P. is 3, 7, 11, 15, ...

Conclusion

Arithmetic Progressions are a crucial part of mathematical sequences, and understanding their properties and formulas is essential for problem-solving. By working through these examples, you can gain a deeper understanding of APs and improve your mathematical skills. Remember to practice regularly and apply these concepts to various problems to master them effectively.