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Understanding nth Term of AP
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Today, we're going to learn about the nth term of an Arithmetic Progression. Does anyone remember what an AP is?
Isn't it a sequence where you add a fixed number to get the next number?
Exactly! Now, if we want to find the term at a specific position, like the fifth term, how do we do that?
We can add the common difference to the first term repeatedly?
That's right, but there's a formula that makes it much quicker! The nth term can be calculated as a_n = a + (n - 1)d. Can anyone tell me what each part means?
I think 'a' is the first term and 'd' is the common difference!
Great job! So if a = 8000 and d = 500 for Reena's salary, how could we find her salary for the 5th year?
It would be βΉ8000 + (5-1) Γ βΉ500, which equals βΉ10000!
Exactly! Let's summarize: the nth term formula helps us quickly find any term in an AP.
Examples of nth Term Calculation
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Now, let's practice with some examples. If I give you a sequence like 2, 7, 12, whatβs the 10th term?
We first find 'a' which is 2, and 'd', which is 5. So, using the formula: a_n = a + (n - 1)d, we would find: 2 + (10 - 1) Γ 5.
Exactly! Can you finish that calculation?
Sure, it would be 2 + 9 Γ 5 = 2 + 45 = 47.
Well done! This shows how easily we can find any term using the formula.
Finding Specific Terms
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Now letβs shift to finding a term, say -81, in the sequence 21, 18, 15. Who can set up the equation for me?
We can use the nth term formula. Here, a = 21, and d = -3. It becomes -81 = 21 + (n - 1)(-3).
Exactly! Now, how would you solve that?
We rearrange to get -81 - 21 = -3n + 3, so -102 = -3n, leading to n = 34!
Perfect! You found that -81 is the 35th term. What if I asked if 0 can ever be a term in the same AP?
If we set up the equation for 0, we could solve and check for n!
Absolutely! Being able to manipulate the nth term formula helps with identifying terms within a sequence.
Real-life Applications of APs
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Arithmetic Progressions are everywhere! Can anyone think of real-world examples?
Salaries and savings could be a good example with increments!
Or even the production of items, like we learned with TV sets!
Excellent! In finance, knowing the nth term helps in projections. Why is understanding AP important in budgeting?
Because it allows us to predict future expenses or savings based on past data!
Exactly! Using the concept of nth terms can simplify planning and forecasting.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, we explore the concept of the nth term in an Arithmetic Progression (AP). By analyzing Reena's salary increment example, we derive the formula for the nth term of an AP, a_n = a + (n-1)d, where 'a' is the first term and 'd' is the common difference. The section also includes several examples showcasing how to apply this formula to determine specific terms in various arithmetic sequences.
Detailed
nth Term of an AP
In this section, we delve into how to find the nth term of an Arithmetic Progression (AP). An AP is a sequence where each term after the first is formed by adding a constant, known as the common difference (d), to the previous term.
Key Points:
- Understanding the nth term: The nth term in an AP can be found using the formula: a_n = a + (n - 1)d, where:
- a is the first term,
- d is the common difference.
- Example Application: Consider the case of Reena, who starts with a monthly salary of βΉ8000 and receives an annual increment of βΉ500. To find her salary in the 5th year, we can compute:
- Salary for the 5th year = a + (5-1)d = βΉ8000 + 4 Γ βΉ500 = βΉ10000.
- Examples and Practice: Further examples illustrate finding specific terms in various sequences, checking if a number is part of an AP, and reverse calculations to derive values like the first term or common difference.
This formula and understanding are crucial in various applications such as financial forecasting, pattern recognition, and more. Mastery of these concepts allows for efficient resolution of complex arithmetic-related problems.
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Understanding the nth Term
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Let us consider the situation again, given in Section 5.1 in which Reena applied for a job and got selected. She has been offered the job with a starting monthly salary of 8000, with an annual increment of 500. What would be her monthly salary for the fifth year?
Detailed Explanation
The key idea here is to calculate how Reena's salary increases over the years. Starting with an initial salary of 8000, and increasing by500 each year means we will add 500 for each year that passes. For instance, in her first year, she earns8000, in the second year, it is 8500 (which is8000 + `500). This process continues, establishing a pattern where each year's salary builds on the last. Hence, we derive a formula for calculating any year's salary efficiently.
Examples & Analogies
Think of this situation like planting a tree where every year the tree grows taller by a fixed amount. If you measure the tree's height in the first year and continue adding that same height every year, at the end of each year, you'll be able to predict how tall the tree will be in the future without having to measure it each time.
Deriving the Formula for the nth Term
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Salary for the 15th year = [8000 + (15 β 1) Γ 500] = 15000.
Detailed Explanation
We observe that by simplifying the salary for any year, moving from 1 to n can be achieved with the formula: Salary(n) = First Salary + (n - 1) * Increment. Hence, when we rearrange our observations mathematically, we create a formula that allows for quick feedback rather than recalculating each year.
Examples & Analogies
Imagine if you are juggling balls, and each time you catch one, you add another ball to the pile youβre juggling. The first year, you catch one ball, then each year you catch one more. If someone asks you how many youβd be juggling after n years, rather than counting each one, you can simply say the initial count plus a ball for each additional year.
General Formula for the nth Term
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So, the nth term a_n of the AP with first term a and common difference d is given by a_n = a + (n β 1) d.
Detailed Explanation
This chunk introduces the standardized formula for the nth term of an arithmetic progression. The first term is labeled 'a', and the difference between each term 'd'. To find any term in the sequence, we calculate it by taking the first term and adding the product of the increment and how far along we are in the sequence minus one.
Examples & Analogies
Consider baking cookiesβif the first batch yields 12 cookies (first term), and every subsequent batch adds 4 more cookies (the increment), the number of cookies youβll have after 'n' batches is easily calculable using this formula without needing to bake batch by batch each time.
Examples of Finding the nth Term
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Example 3: Find the 10th term of the AP: 2, 7, 12, . . . Solution: Here, a = 2, d = 7 - 2 = 5 and n = 10. We have a_n = 2 + (10 - 1) Γ 5 = 2 + 45 = 47. Therefore, the 10th term of the given AP is 47.
Detailed Explanation
In this example, we first identify the first term 'a' and the common difference 'd'. Using the nth term formula, we substitute the values for 'n', 'a' and 'd'. Spotting how the components fit together helps us find our target term, the 10th term, efficiently.
Examples & Analogies
Similar to filling a jar with marbles where the first addition is 2 marbles and thereafter 5 marbles per time. By counting how many intervals (n) of additions you've completed using our formula, you can quickly know how many marbles you have without directly counting each time.
Negative Terms in AP
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Example 4: Which term of the AP: 21, 18, 15, . . . is β81? Also, is any term 0? Give reason for your answer.
Detailed Explanation
In this scenario, we need to find out which term matches with -81. By using the nth term formula, we can set -81 equal to a_n, and solve for 'n' to find its position in the sequence. Similarly, checking if 0 appears requires checking if there exists an integer 'n' that satisfies the nth term equation leading to 0.
Examples & Analogies
Think of this like finding the day of the month that falls below a certain threshold. In this case, we are reversing the logic of counting days, instead counting back to a point of relevance, which allows for efficient solving.
Determining an AP from Given Information
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Example 5: Determine the AP whose 3rd term is 5 and the 7th term is 9.
Detailed Explanation
This example illustrates how to work backwards from known terms to derive 'a' and 'd'. By setting up equations based on the nth term definitions for the 3rd and 7th terms, we can solve them simultaneously to find the sequence's first term and common difference.
Examples & Analogies
This is like piecing together a jigsaw puzzle where we use the pieces we know (specific terms) to find the edges (first term and common difference) of the puzzle, leading us to the complete picture of the AP.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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nth Term: Formula for nth term is a_n = a + (n - 1)d.
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Common Difference: The difference, d, that remains constant between terms.
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First Term: The starting point of the AP denoted as 'a'.
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The Importance of APs: Used in practical financial planning and budgeting.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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If a = 2 and d = 5, then 10th term = 2 + (10 - 1) * 5 = 47.
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For the AP 4, 7, 10, find the 7th term: a = 4, d = 3; hence a_7 = 4 + (7-1)*3 = 22.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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To find the nth term, just take a, add d times (n minus one), thatβs how itβs done.
π Fascinating Stories
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Imagine a bank where you're depositing a fixed amount each month; your balance grows steadily, just like terms in an AP!
π§ Other Memory Gems
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Remember A for Arithmetic, D for Difference to find the nth term!
π― Super Acronyms
AP = (A + (n-1)D) for finding terms swiftly.
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 any two consecutive terms is constant.
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Term: Common Difference (d)
Definition:
The fixed amount added to each term in an Arithmetic Progression to obtain the next term.
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Term: First Term (a)
Definition:
The initial term in an Arithmetic Progression.
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Term: nth Term (a_n)
Definition:
The term in the sequence of an AP that corresponds to the index n.
Example 2
In a flower bed, there are 30 tulip plants in the first row, 28 in the second, 26 in the third, and so on. There are 10 plants in the last row. How many rows are there in the flower bed?
Solution: The number of tulip plants in the 1st, 2nd, 3rd, ..., rows are:
\[ 30, 28, 26, \ldots \]
It forms an AP (Arithmetic Progression) (Why?). Let the number of rows in the flower bed be \( n \).
Then,
\[ a = 30, \, d = 28 - 30 = -2, \, a_n = 10 \]
As,
\[ a_n = a + (n - 1) d \]
We have,
\[ 10 = 30 + (n - 1)(-2) \]
\[ 10 = 30 - 2(n - 1) \]
\[ 10 = 30 - 2n + 2 \]
\[ 2n = 30 + 2 - 10 \]
\[ 2n = 22 \]
\[ n = 11 \]
i.e., So there are 11 rows in the flower bed.