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Introduction to Arithmetic Progressions
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Alright class, today weβre going to learn about Arithmetic Progressions, or AP for short. Can anyone tell me what they think an AP might be?
Is it a kind of pattern in numbers?
Exactly! In an AP, each term after the first is formed by adding a fixed number, called the common difference, to the previous term. For example, if we start with 2 and add 3, our sequence will be 2, 5, 8, 11, and so on.
What if the common difference is negative?
Great question! If the common difference is negative, the terms will decrease. For instance, starting at 10 and subtracting 2 gives us 10, 8, 6, 4. In both cases, we have an arithmetic progression.
So the first term is 'a' and the common difference is 'd', right?
Correct! Remember, we're denoting the first term as 'a', and the common difference as 'd'. Can anyone find the nth term formula?
Is it a_n = a + (n - 1)d?
Spot on! Letβs keep this in mind as we move forward.
Finding Common Difference
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Now that we know what an AP is, how can we check if a list of numbers forms an AP?
By finding the differences between them?
Exactly! If the difference between consecutive terms is constant, then we have an AP. For example, if we examine the numbers 3, 7, 11, 15, whatβs the common difference here?
Itβs 4!
Thatβs right. Each term increases by 4, making this list an AP. Now, what about the list 1, 1, 1, 1?
Thatβs also an AP because the common difference is 0.
Exactly! Any series where every term is the same forms an AP.
Sum of n Terms of an AP
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Next, letβs talk about how we can find the sum of the first n terms of an AP. Who remembers the sum formula?
Is it S_n = n/2 [2a + (n-1)d]?
Excellent! This formula is really useful. Letβs consider the AP 2, 4, 6, 8. If we want to find the sum of the first 4 terms, how can we use this formula?
So, a is 2, and d is 2, and n is 4?
Thatβs correct. Plug those values into the formula. What do you get?
S_4 = 4/2 [2 * 2 + (4 β 1)*2] = 2 [4 + 6] = 2 * 10 = 20!
Great job! This method helps us sum APs efficiently.
Real-Life Applications of APs
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Can anyone think of real-life situations where we can see Arithmetic Progressions?
Salary increments! Like when someone gets a fixed raise every year.
Exactly! Another example could be distances in athletics, where every lap is the same distance. These are patterns of AP.
Are all salary increments APs, though?
Good point! Not necessarily, if they vary over time, that would not be an AP.
So, if the increments are consistent, then yes!
Exactly! Arithmetic Progressions are all around us if we look closely.
Introduction & Overview
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Quick Overview
Standard
The section explains what Arithmetic Progressions are, including their definition, common difference, and how to derive their nth term and sum of the first n terms. It also provides practical examples and applications in real-life scenarios.
Detailed
Detailed Summary
Arithmetic Progressions (AP) are sequences of numbers where the difference between consecutive terms is constant, referred to as the common difference (d). The section emphasizes that the first term is denoted as 'a' and the nth term can be calculated using the formula:
$$a_n = a + (n-1)d$$
Examples such as salary increment patterns and physical measurements illustrate AP in real life. The chapter further explores identifying whether a sequence is an AP by checking if the differences between consecutive terms are equal. The section concludes with how to find the sum of the first n terms in an AP using the formula:
$$S_n = \frac{n}{2} [2a + (n-1)d]$$
This foundational knowledge is critical for solving various mathematical and real-world problems involving sequences.
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Introduction to Arithmetic Progressions
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Consider the following lists of numbers :
(i) 1, 2, 3, 4, . . .
(ii) 100, 70, 40, 10, . . .
(iii) β3, β2, β1, 0, . . .
(iv) 3, 3, 3, 3, . . .
(v) β1.0, β1.5, β2.0, β2.5, . . .
Each of the numbers in the list is called a term. Given a term, can you write the next term in each of the lists above? If so, how will you write it? Perhaps by following a pattern or rule. Let us observe and write the rule.
Detailed Explanation
An Arithmetic Progression (AP) consists of a sequence of numbers where the difference between consecutive terms is constant. This fixed difference is called the common difference, denoted as 'd'. Each list presented shows a pattern in which you can identify the rule to find the next term.
- For the list (i), each term increases by 1, so it's 1 + 1 = 2, and so forth.
- In list (ii), each term decreases by 30, leading to 100 - 30 = 70.
- List (iii) illustrates that each term increases by 1 starting from -3, and so on for the rest. The same concept applies to lists (iv) and (v). All these examples depict an arithmetic sequence, highlighting that when moving from one term to the next, you can determine future terms using the common difference.
Examples & Analogies
Think of climbing stairs. Each step is equidistant apart, similar to how terms in an arithmetic sequence are spaced by a consistent amount, the common difference. Just like knowing the height of the first step allows you to easily find the height of the following steps, knowing the first term and the common difference helps you find subsequent terms in an AP.
Definition and Characteristics of AP
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In all the lists above, we see that successive terms are obtained by adding a fixed number to the preceding terms. Such list of numbers is said to form an Arithmetic Progression (AP).
So, an arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term. This fixed number is called the common difference of the AP. Remember that it can be positive, negative or zero.
Detailed Explanation
An Arithmetic Progression is fundamentally a sequence where consecutive terms are generated by adding the same value, known as the common difference (d), to the previous term. This can manifest in various forms:
- If d is positive, the terms increase.
- If d is negative, the terms decrease.
- If d equals zero, all terms remain the same.
For example, if the first term (a) is 5 and d is 3, the sequence goes 5, 8, 11, 14,... If d is -2, starting from 5 yields 5, 3, 1, -1,... This emphasizes the versatility of APs.
Examples & Analogies
Consider a savings plan where you deposit a fixed amount every month. If you deposit $50 each month, your account balance follows an arithmetic progression: $50, $100, $150, $200, and so on. The common difference is the amount you add, which helps you predict your total savings at any point in time.
Identification of AP
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Let us denote the first term of an AP by a , second term by a , . . ., nth term by a and the common difference by d. Then the AP becomes a , a , a , . . ., a . So, a β a = a β a = . . . = a β a = d.
Detailed Explanation
To denote the terms in an arithmetic progression clearly, we label the first term as aβ, the second term as aβ, and the nth term as aβ. The common difference d can be identified between any two consecutive terms:
- d = aβ - aβ
- d = aβ - aβ
- This pattern continues for any terms in the sequence. This relationship illustrates how each term can be derived from its predecessor using the common difference.
Examples & Analogies
Imagine organizing a baking schedule where every week you make three more cookies than the previous week. If your first week you bake 10 cookies, your second week would be 10 + 3 = 13, leading to a clear AP: 10, 13, 16, 19,... Here, 10 is your first term, and 3 is the common difference.
Examples of Arithmetic Progressions
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Some more examples of AP are:
(a) The heights ( in cm ) of some students of a school standing in a queue in the morning assembly are 147 , 148, 149, . . ., 157.
(b) The minimum temperatures ( in degree celsius ) recorded for a week in the month of January in a city, arranged in ascending order are β 3.1, β 3.0, β 2.9, β 2.8, β 2.7, β 2.6, β 2.5
(c) The balance money ( in ) after paying 5 % of the total loan of 1000 every month is 950, 900, 850, 800, . . ., 50.
(d) The cash prizes ( in ) given by a school to the toppers of Classes I to XII are, respectively, 200, 250, 300, 350, . . ., 750.
(e) The total savings (in) after every month for 10 months when ` 50 are saved each month are 50, 100, 150, 200, 250, 300, 350, 400, 450, 500.
Detailed Explanation
In the examples listed, each scenario shows a distinct arithmetic progression where a fixed difference contributes to the creation of subsequent terms:
- (a) Height increases by 1 cm for each student.
- (b) Temperature rises by 0.1Β°C each day.
- (c) The loan balance decreases by 50 βΉ each month.
- (d) Each cash prize increases by 50 βΉ with each class.
- (e) Savings increase by a steady amount of 50 βΉ monthly. Understanding these examples provides insight into various real-world applications of AP.
Examples & Analogies
Picture a hotel with a rising price per night where the cost increases by a fixed amount (such as $20) each week. If the price starts at $100 in the first week, it becomes $120 in the second, $140 in the third, and so on. This scenario represents an arithmetic progression where you can easily predict how much a stay will cost in subsequent weeks!
Finding the Common Difference
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It is left as an exercise for you to explain why each of the lists above is an AP. You can see that a, a + d, a + 2d, a + 3d, . . . represents an arithmetic progression where a is the first term and d the common difference. This is called the general form of an AP.
Detailed Explanation
The definition of an arithmetic progression can be summarized into a formula: a, a + d, a + 2d, a + 3d,... where a is the initial term, and d is the consistent difference that can be either positive, negative, or zero. By analyzing the lists presented, you can see how to apply the definition and recognize whether a number sequence is an AP based on this pattern.
Examples & Analogies
Think of the layers of cake in a tiered cake: if you add a consistent height for each layer, the overall structure becomes an arithmetic progression by height. If each layer is 2 inches taller than the one beneath, starting at 4 inches, your layers are 4, 6, 8, 10,... consistently rising as an arithmetic progression!
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Arithmetic Progression: A sequence formed by repeated addition of a fixed number.
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Common Difference: The consistent difference between consecutive terms.
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Formula for nth term: a_n = a + (n-1)d
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Sum of n terms: S_n = n/2 [2a + (n-1)d]
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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In a salary increment scenario, if a person earns $1000 initially and receives a $100 increment yearly, the sequence of salaries forms an AP: 1000, 1100, 1200, ...
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A ladder where the distance between rungs decreases consistently can be seen as an AP.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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In an AP, terms don't stray, they grow by d each day.
π Fascinating Stories
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Imagine climbing a staircase with equally spaced steps, each step higher represents the addition of the common difference.
π§ Other Memory Gems
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Remember: AAP - Always Add the Progression (A for 'Arithmetic', A for 'Add', P for 'Progression').
π― Super Acronyms
AP = Addition of Progression.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Arithmetic Progression (AP)
Definition:
A sequence of numbers in which the difference between consecutive terms is constant.
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Term: Common Difference (d)
Definition:
The fixed amount added to each term to get the next term in an AP.
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Term: First Term (a)
Definition:
The initial term in an arithmetic progression.
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Term: nth Term
Definition:
The term which is in the position n in a sequence.
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Term: Sum of First n Terms (S_n)
Definition:
The sum of the first n terms in an arithmetic progression.