Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take mock test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Finding the 25th Term
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Today, we're going to focus on how to find specific terms in an arithmetic sequence. Let's take the sequence 4, 9, 14, 19,... Can anyone tell me the first term and the common difference?
The first term is 4, and the common difference is 5.
Exactly! Now, who can remind the class of the formula for the nth term?
It's T_n = a + (n - 1)d!
Great! So, if we want to find the 25th term, what values do we plug into the formula?
We use a = 4, d = 5, and n = 25.
Correct! Now, let's calculate it together. I want everyone to notice how we can apply this formula to any arithmetic sequence. The 25th term will be T_25 = 4 + (25 - 1) * 5.
That equals 4 + 120, which is 124!
Well done! So, the 25th term is 124. Remember to practice this formula with different sequences.
Calculating the Sum of First n Terms
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Now let's move on to calculating the sum of the first n terms. Can anyone recall the sum formula?
It's S_n = n/2 * (2a + (n - 1)d) or S_n = n/2 * (a + l)!
Excellent! Let’s work on finding the sum of the first 30 terms of the sequence 10, 8, 6, … What's a and d here?
a is 10, and d is -2.
Exactly. So how many terms do we have?
30 terms!
Great! Plug those into the sum formula, what do you get?
Using the first formula, S_30 = 30/2 * (2*10 + (30 - 1)(-2)).
That's right, now compute it.
So, S_30 = 15 * (20 - 58) = 15 * -38 = -570.
Perfect! Remember, in a decreasing sequence like this, the sum can also be negative!
Application Problem Discussion
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Let’s discuss an application problem. A person saves $100 in the first month, then $120 in the second month. What do we want to find here?
We need to find out how much they saved after 12 months.
Correct! What is the sequence for their savings?
It starts at $100 and goes up by $20 each month.
Exactly! So what do we know? Can we determine a, d, and n?
a = 100, d = 20, and n = 12!
Now use the sum formula to find S_n.
Okay, so S_12 = 12/2 * (2*100 + (12-1)*20). That’s 6 * (200 + 220), which is 6 * 420 = 2520.
Exactly right! $2520 saved in total after 12 months. Well done!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The practice questions challenge students on various aspects of arithmetic sequences, including finding specific terms, calculating sums, and solving application problems. These exercises reinforce the knowledge gained throughout the chapter.
Detailed
The 'Practice Questions' section consists of five carefully crafted questions that span key concepts of arithmetic sequences. Students will engage with scenarios requiring them to find specific terms of a sequence, calculate the sum of the first n terms, and tackle real-world applications of arithmetic progressions. This section aims to enhance understanding through hands-on practice, ensuring students can confidently apply the learned concepts.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Question 1: Finding the 25th Term
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
- Find the 25th term of the sequence: 4, 9, 14, 19, ...
Detailed Explanation
This question asks us to find the 25th term of the arithmetic sequence where the first few terms are 4, 9, 14, and 19. To solve this, we first identify the first term (a) and the common difference (d). Here, a = 4 and the common difference is d = 9 - 4 = 5. We can use the nth term formula for arithmetic sequences: \( T_n = a + (n - 1)d \). Substituting the known values for n = 25 gives us: \( T_{25} = 4 + (25 - 1) * 5 = 4 + 120 = 124 \). Thus, the 25th term is 124.
Examples & Analogies
Imagine counting the number of steps on a staircase where each step adds 5 more steps than the previous one, starting from 4 steps. The 25th step would represent how many total steps you'd count after climbing that many steps.
Question 2: Total Number of Terms
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
- How many terms are there in the sequence: 7, 13, 19, ..., 121?
Detailed Explanation
In this question, we need to determine how many terms are in the sequence that starts at 7 and increases in each step by 6 (the common difference d). The last term of the sequence is 121. We use the formula for the nth term of an arithmetic sequence: \( T_n = a + (n - 1)d \). We set \( T_n = 121 \), with a = 7 and d = 6: \( 121 = 7 + (n - 1) * 6 \). Solving for n gives us: \( 121 - 7 = (n - 1) * 6 \), which simplifies to 114 = (n - 1) * 6. Therefore, n - 1 = 19, meaning n = 20. Thus, there are 20 terms.
Examples & Analogies
Think of a row of chairs where the first chair has 7 students and each subsequent chair has 6 more than the previous chair. If the last chair holds 121 students, finding out how many chairs you have in total is similar to figuring out the total number of terms in this sequence.
Question 3: Sum of the First 30 Terms
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
- Find the sum of the first 30 terms of the sequence: 10, 8, 6, ...
Detailed Explanation
Here, we are asked to find the total of the first 30 terms in the sequence starting at 10 and decreasing by 2 (common difference d = -2). To find this sum, we use the formula for the sum of the first n terms of an arithmetic sequence: \( S_n = \frac{n}{2} (2a + (n - 1)d) \). With n = 30, a = 10, and d = -2, we calculate: \( S_{30} = \frac{30}{2} (2 * 10 + (30 - 1)(-2)) = 15 (20 - 58) = 15 * -38 = -570 \). Therefore, the sum of the first 30 terms is -570.
Examples & Analogies
Imagine a budget that starts at $10 but each week loses $2 until you've accounted for 30 weeks. You might end up with a total loss, illustrating how the sum of terms adds up in an arithmetic sequence!
Question 4: Finding First Term and Common Difference
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
- The 5th term of an arithmetic sequence is 20 and the 12th term is 41. Find the first term and the common difference.
Detailed Explanation
In this problem, we need to find the first term (a) and the common difference (d) of an arithmetic sequence given the 5th term (20) and the 12th term (41). We use the formula for the nth term to set up two equations: \( T_5 = a + 4d = 20 \) and \( T_{12} = a + 11d = 41 \). Solving these simultaneously, we can rewrite the first equation as a = 20 - 4d. Substituting this into the second equation gives us: \( (20 - 4d) + 11d = 41 \). Solving for d yields d = 3. Then substituting back into the first equation gives a = 20 - 12 = 8. The first term is 8 and the common difference is 3.
Examples & Analogies
Consider a game where after every 4 plays your score is 20, and after 11 plays, it’s 41. Finding out your initial score and how much each play increases is like figuring out the first term and common difference in your sequence.
Question 5: Total Seats in a Stadium
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
- A stadium has 25 rows. The first row has 20 seats, and each successive row has 2 more seats than the previous one. How many seats are there in total?
Detailed Explanation
In this scenario, we are determining the total number of seats in a stadium with 25 rows where the first row has 20 seats and each subsequent row increases by 2 seats (common difference d = 2). The number of seats in the nth row can be described with the formula: \( T_n = a + (n - 1)d \). Here a = 20, and we want T_25. This gives: \( T_{25} = 20 + (25 - 1) * 2 = 20 + 48 = 68 \). To find the total number of seats, we calculate the sum of all rows: \( S_n = \frac{n}{2}(a + l) \). Here, n = 25, a = 20, and l (last term) = 68. Thus, total seats = \( S_{25} = \frac{25}{2}(20 + 68) = 25 * 44 = 1100 \). There are a total of 1100 seats.
Examples & Analogies
Imagine setting up a concert in a stadium where each row fills up with a few more chairs than the one before. Calculating how many chairs in total helps visualize the attendance for a big event!
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Arithmetic Sequence: A sequence with a constant difference between terms.
-
Common Difference: The fixed difference between consecutive terms.
-
nth Term Formula: T_n = a + (n - 1)d to find any term.
-
Sum Formula: S_n = n/2 * (2a + (n - 1)d) for calculating the sum.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Finding the 25th term of the sequence 4, 9, 14, 19 using the nth term formula.
-
Calculating the sum of the first 30 terms of the sequence 10, 8, 6, using the sum formula.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
When you find a term, just remember the rule, T_n equals a plus d times n minus one - that’s cool!
📖 Fascinating Stories
-
Imagine a person saving money, starting with $100 and increasing their savings monthly; it's like climbing a staircase, where each step is their saving growth.
🧠 Other Memory Gems
-
To remember the formula for the sum, think of S_n = n over 2 times (2a plus (n minus 1)d) - the last part is simple if you know your n's!
🎯 Super Acronyms
To remember terms in an arithmetic sequence
- 'A's for Arithmetic
- 'C' for Common difference
- 'n' for number of terms
- 'S' for Sum.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Arithmetic Sequence
Definition:
A sequence of numbers in which the difference between consecutive terms is constant.
-
Term: Common Difference (d)
Definition:
The constant difference between consecutive terms in an arithmetic sequence.
-
Term: nth Term (T_n)
Definition:
The term located at position n in an arithmetic sequence, calculated using T_n = a + (n - 1)d.
-
Term: Sum of First n Terms (S_n)
Definition:
The total of the first n terms in an arithmetic sequence.