Arithmetic Sequence (Explicit Formula)

Summary

This video explains the concepts of arithmetic sequences, focusing on the distinction between sequences and series. It covers how to identify the common difference, denoted as D, which is added to each term to generate the next. The video illustrates how to label terms in a sequence using both N values and subscript notation. It also introduces the explicit formula for finding any term in the sequence, showing how to calculate specific terms like the sixth or hundredth term. Overall, it provides a clear foundation for understanding arithmetic sequences and their properties.

Takeaways

• 😀 A sequence is a list of numbers, while a series is the sum of those numbers.
• 😀 In an arithmetic sequence, a constant value, known as the common difference (D), is added to each term to get to the next term.
• 😀 The common difference can be calculated by subtracting consecutive terms (e.g., 21 - 17 = 4).
• 😀 The notation N = 1, 2, 3, etc., refers to the position of terms in the sequence (first term, second term, etc.).
• 😀 The notation a sub n (e.g., a_1, a_2, a_3) indicates the value of a specific term in the sequence.
• 😀 To find the nth term in an arithmetic sequence, use the formula a_n = a_1 + D × (n - 1).
• 😀 The first term (a_1) is critical in determining the values of subsequent terms.
• 😀 When calculating the nth term, you only add the common difference (D) a number of times equal to (n - 1).
• 😀 The explicit formula derived for this example is a_n = 13 + 4n, simplifying the calculation of any term in the sequence.
• 😀 By substituting different values of n into the formula, any term can be quickly calculated, such as the hundredth term.

Q & A

What is the difference between a sequence and a series?

A sequence is a list of numbers, while a series is the sum of those numbers.

What does 'arithmetic' refer to in the context of sequences?

In the context of sequences, 'arithmetic' refers to adding the same value, known as the common difference, to get to the next term.

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

The common difference (D) is the constant value added to each term to reach the next term in the sequence.

How do you calculate the common difference in a sequence?

To calculate the common difference, subtract the previous term from the current term (e.g., 21 - 17 = 4).

What does N represent in the context of sequences?

N represents the term number in the sequence, indicating its position (e.g., N=1 for the first term).

What does the notation 'a_sub_n' represent?

'a_sub_n' represents the value of the nth term in a sequence.

How do you find the sixth term of an arithmetic sequence starting at 17 with a common difference of 4?

To find the sixth term, start at 17 and add the common difference (4) five times, resulting in 37.

What is the explicit formula for finding any term in an arithmetic sequence?

The explicit formula is a_sub_n = a_sub_1 + D * (n - 1), where a_sub_1 is the first term, D is the common difference, and n is the term number.

How can you simplify the explicit formula for the sequence mentioned in the transcript?

By substituting the values from the transcript, the simplified formula becomes a_sub_n = 13 + 4n.

How can the explicit formula be used to find the hundredth term in the sequence?

To find the hundredth term, substitute n with 100 in the formula: a_sub_100 = 13 + 4 * 100, which equals 413.