Functions 7.1 - Arithmetic Sequences

Summary

This lesson introduces arithmetic sequences and the concept of discrete functions, offering a fresh approach for students struggling with functions. The video explains how each term in an arithmetic sequence is found by adding or subtracting a constant difference, with examples on finding the general term. The instructor demonstrates solving for terms in sequences using logical steps, including techniques like elimination and recursive formulas. Students are guided through practice problems to reinforce understanding and gain confidence in applying these principles. The video emphasizes the importance of grasping these foundational concepts for success in more advanced topics.

Takeaways

• 😀 Arithmetic sequences involve a constant difference (D) between consecutive terms, which can be positive or negative.
• 😀 The general formula for the nth term in an arithmetic sequence is: T_n = a + (n - 1) * D, where 'a' is the first term, 'D' is the common difference, and 'n' is the position of the term.
• 😀 To find the common difference (D), subtract any two consecutive terms in the sequence.
• 😀 The recursive formula for an arithmetic sequence expresses each term in relation to the previous term: T_n = T_(n-1) + D.
• 😀 The nth term formula allows you to calculate any term in the sequence, including large values like T_100 or T_1000.
• 😀 To solve for an unknown term when given two terms in the sequence, calculate the common difference and then use the general formula to find the required term.
• 😀 When the first term (a) and common difference (D) are unknown, you can use two known terms in the sequence to find them using a system of equations.
• 😀 The difference between any two terms divided by the difference in their positions helps calculate the common difference (D).
• 😀 Recursive formulas require you to know the first term and the relationship between consecutive terms to find subsequent terms.
• 😀 To find the total number of terms in a sequence, use the general term formula and solve for 'n' when the last term (T_n) is known.

Q & A

What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers where each term is found by adding or subtracting the same value (called the common difference) to the previous term.

What does the common difference (D) in an arithmetic sequence represent?

The common difference (D) represents the constant amount added or subtracted from one term to the next in the sequence. It can be positive (if terms increase) or negative (if terms decrease).

How is the nth term of an arithmetic sequence determined?

The nth term of an arithmetic sequence is determined using the formula: T_n = a + (n - 1) × D, where 'a' is the first term, 'n' is the term number, and 'D' is the common difference.

In the sequence 2, 9, 16, ..., how would you find the general term?

In this sequence, the first term is 2 (a = 2) and the common difference is 7 (D = 7). Using the general term formula, T_n = 2 + (n - 1) × 7, the general term is T_n = 7n - 5.

What does the recursive formula for a sequence involve?

The recursive formula for a sequence defines each term based on the previous term. It expresses T_n in terms of T_(n-1) and includes an initial condition, typically stating the first term.

How do you derive the recursive formula for the sequence 32, 26, 20, 14, ...?

For the sequence 32, 26, 20, 14, ..., the common difference is -6. The recursive formula is T_n = T_(n-1) - 6, with the first term T_1 = 32.

What is the general term for the sequence 7, 2, -3, -8, ...?

For the sequence 7, 2, -3, -8, ..., the first term is 7 (a = 7) and the common difference is -5 (D = -5). The general term is T_n = 12 - 5n.

In the problem where T_7 = 53 and T_11 = 97, how do you find the 100th term?

To find the 100th term, first calculate the common difference. The difference between T_7 and T_11 is 44, which is the result of adding the common difference 4 times. Therefore, D = 44 / 4 = 11. Using T_11 = 97, calculate T_100 by adding the common difference 89 times: T_100 = 97 + 11 × 89 = 1076.

How do you determine the number of terms in the sequence -10, -14, -18, ..., where the last term is -138?

To determine the number of terms, first find the common difference, which is -4 (from -10 to -14). Then, use the formula for the nth term: T_n = a + (n - 1) × D. Substituting a = -10, D = -4, and T_n = -138, solve for n: -138 = -10 + (n - 1) × (-4). Solving gives n = 33, so there are 33 terms in the sequence.

What is the method for solving a system of equations to find the general term when the first term and common difference are unknown?

When given two terms in an arithmetic sequence (such as T_7 and T_11), set up a system of equations. Use the general term formula T_n = a + (n - 1) × D for both terms. Subtract the equations to eliminate 'a' and solve for D. Once D is found, substitute it back into one of the equations to solve for 'a'. Then, you can form the general term.