The 𝑛th Term Formula - 2 | 13. Arithmetic Sequences | IB Class 10 Mathematics – Group 5, Algebra
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2 - The 𝑛th Term Formula

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Understanding the 𝑛th Term Formula

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Teacher
Teacher

Today, we're diving into the 𝑛th term formula for arithmetic sequences, which is a powerful way to find any term in a sequence without listing them all. The formula is: 𝑇 = π‘Ž + (π‘›βˆ’1)𝑑. Can anyone tell me what the letters in this formula represent?

Student 1
Student 1

I think π‘Ž is the first term!

Teacher
Teacher

That's correct, Student_1! And what about 𝑑?

Student 2
Student 2

Isn't 𝑑 the common difference between the terms?

Teacher
Teacher

Exactly! Now, who can tell me what 𝑛 represents?

Student 3
Student 3

It’s the position of the term we want to find, right?

Teacher
Teacher

Well done, Student_3! To make it easier to remember, think of the phrase 'A Difference in Term Position', where each word reminds us of π‘Ž, 𝑑, and 𝑛. Let's practice using this formula!

Applying the 𝑛th Term Formula

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Teacher
Teacher

Now, let’s find the 10th term of the sequence: 3, 7, 11, 15. Who remembers how to identify the values of π‘Ž, 𝑑, and 𝑛?

Student 4
Student 4

I think π‘Ž is 3, and 𝑑 is 4, but I'm not sure how to set 𝑛.

Teacher
Teacher

Great, Student_4! Since we're looking for the 10th term, 𝑛 will be 10. Now, let's plug these values into our formula. What do we get?

Student 2
Student 2

We would calculate 𝑇 = 3 + (10 - 1) * 4.

Teacher
Teacher

Exactly! And then what is 𝑇?

Student 1
Student 1

That would be 3 + 36, which equals 39!

Teacher
Teacher

Fantastic! So the 10th term is indeed 39. This connection between basic sequences and algebra is very useful. Can anyone think of a situation where knowing a specific term could be helpful?

Practical Applications of the 𝑛th Term Formula

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Teacher
Teacher

Let’s apply our knowledge of the 𝑛th term formula to real-life situations. Can anyone suggest a scenario where we might need to calculate a term in a sequence?

Student 3
Student 3

What if we're saving money each month? Like saving $100, then $120, and $140...

Teacher
Teacher

Excellent example, Student_3! We can think of this as an arithmetic sequence where the first amount saved is 100 and the common difference is 20. If we want to find out how much was saved in the 12th month, how would we set up our formula?

Student 4
Student 4

So it would be π‘Ž = 100, 𝑑 = 20, and 𝑛 = 12?

Teacher
Teacher

Spot on! Now plug those values into the formula. What do you calculate?

Student 2
Student 2

It would be 100 + (12 - 1) * 20, which gives us $320 saved in the 12th month.

Teacher
Teacher

Exactly, $320! This shows just how useful the 𝑛th term formula can be in everyday life.

Introduction & Overview

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Quick Overview

This section introduces the 𝑛th term formula for arithmetic sequences, allowing students to calculate any term based on the first term and common difference.

Standard

In this section, the 𝑛th term formula is presented, helping students find specific terms in an arithmetic sequence. The formula, 𝑇 = π‘Ž + (π‘›βˆ’1)𝑑, is explained with an example, demonstrating its importance in identifying terms efficiently.

Detailed

In-Depth Summary

The 𝑛th term formula is a crucial tool in understanding arithmetic sequences, expressed as:

𝑇 = π‘Ž + (π‘›βˆ’1)𝑑

Where:
- 𝑇 is the 𝑛th term,
- π‘Ž is the first term,
- 𝑑 is the common difference,
- 𝑛 is the term position.

This formula allows students to determine the value of any term in an arithmetic sequence efficiently. For instance, if we take the sequence 3, 7, 11, 15, we can find the 10th term by substituting the values into the formula: π‘Ž = 3, 𝑑 = 4 (calculated as 7 - 3), and 𝑛 = 10. By following the formula, we find that the 10th term is 39. This section emphasizes the practical application of the 𝑛th term formula, preparing students for more advanced concepts in algebra and mathematics.

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Introduction to the 𝑛th Term Formula

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To find the value of any term in an arithmetic sequence, use:

𝑇ₙ = π‘Ž + (π‘›βˆ’1)𝑑

Where:
β€’ 𝑇ₙ is the 𝑛th term,
β€’ π‘Ž is the first term,
β€’ 𝑑 is the common difference,
β€’ 𝑛 is the term position.

Detailed Explanation

The 𝑛th term formula provides a way to find any term in an arithmetic sequence without needing to list all the terms. In this formula:
1. 𝑇ₙ represents the term at position 𝑛 in the sequence.
2. π‘Ž denotes the first term in the sequence. This is where the sequence starts.
3. 𝑑 refers to the common difference between consecutive terms. This is the amount added to each term to get to the next term.
4. 𝑛 is the position of the term you want to find.
Thus, to find the value of the term in any sequence, you take the first term, add the product of the common difference and the term position minus one.

Examples & Analogies

Imagine a staircase where each step is evenly spaced apart. The first step represents the first term (π‘Ž), and the distance between each step is the common difference (𝑑). If you want to find out how high the 10th step (𝑇ₙ) is from the ground, you could use the 𝑛th term formula: start from the ground (first step), add the height of each additional step until you reach the 10th step.

Example of Finding the 10th Term

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βœ… Example 1:
Find the 10th term of the sequence: 3, 7, 11, 15, …

Solution:
β€’ π‘Ž = 3
β€’ 𝑑 = 7βˆ’3 = 4
β€’ 𝑛 = 10

𝑇₁₀ = 3+(10βˆ’1)β‹…4 = 3 + 36 = 39.

Detailed Explanation

In this example, we calculate the 10th term of the arithmetic sequence:
1. Identify the first term (π‘Ž): Here, π‘Ž = 3.
2. Calculate the common difference (𝑑): Between 3 and 7, the difference is 4 (7βˆ’3).
3. Select the term position (𝑛): We're looking for the 10th term, so 𝑛 = 10.
4. Plug these values into the 𝑛th term formula: 𝑇₁₀ = 3 + (10βˆ’1)β‹…4 = 3 + 36 = 39. Thus, the 10th term is 39.

Examples & Analogies

Think of this like a factory producing widgets. The first widget comes off the line at unit 3, and each subsequent widget comes off 4 units later. If you want to know when the 10th widget will be completed, you can calculate it using the formula. By the time you count up to the 10th widget, it turns out it will complete at unit 39.

Definitions & Key Concepts

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Key Concepts

  • 𝑇 = π‘Ž + (π‘›βˆ’1)𝑑: The formula to find the 𝑛th term in an arithmetic sequence.

  • Common Difference (𝑑): The constant difference between successive terms in the sequence.

  • First Term (π‘Ž): The starting point of the sequence.

  • Term Position (𝑛): Indicates the position of the term you wish to find.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example: Find the 10th term of the sequence 3, 7, 11, 15 using the formula 𝑇 = 3 + (10-1) * 4 = 39.

  • Example: For a savings plan where a person saves $100, then $120, then $140, find the amount saved in the 12th month using the formula.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎡 Rhymes Time

  • To find the term of nth place, just add a difference to first base.

πŸ“– Fascinating Stories

  • Imagine a person climbing stairs where each step is a fixed height. If they know how high the first step is and how high each step increases, they can find out how high they are on any step!

🧠 Other Memory Gems

  • For the formula, remember: 'A Dog Near' (π‘Ž, 𝑑, 𝑛).

🎯 Super Acronyms

Use the acronym 'TAN' for Term, Add (common difference), N for position.

Flash Cards

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Glossary of Terms

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  • Term: Arithmetic Sequence

    Definition:

    A sequence of numbers with a constant difference between consecutive terms.

  • Term: Common Difference (𝑑)

    Definition:

    The difference between any two consecutive terms in an arithmetic sequence.

  • Term: First Term (π‘Ž)

    Definition:

    The initial term in an arithmetic sequence.

  • Term: 𝑛th Term

    Definition:

    The term at position 𝑛 in a sequence, calculated using the 𝑛th term formula.