Problem 44 - Precalculus with Limits

Question

In Exercises 43 - 48, find a formula for the sum of the first \( n \) terms of the sequence. \( 25, 22, 19, 16, \cdots \)

Step-by-Step Solution

Answer

The formula for the sum of the first n terms of the provided arithmetic sequence is \(S_n = (n/2)(50 - 3(n-1))\)

Step 1: Identify the sequence type and its characteristics

In this case, the given sequence is an arithmetic sequence where the common difference, denoted as 'd', is -3 and the first term, denoted as 'a_1', is 25.

Step 2: Find the nth term formula

Since the arithmetic sequence formula is given by \(a_n = a_1 + (n-1)d\), substitute the values of a_1 and d into this equation to get the nth term of the sequence.

Step 3: Formulate the sum of the arithmetic sequence formula

The sum formula for arithmetic sequence is \(S_n = (n/2)(a_1 + a_n)\). Then, we can replace \(a_n\) in the sum formula with the nth term formula to get \(S_n = (n/2)(2a_1 + (n-1)d)\). Substituting the values of a_1 and d into this equation, we get the sum formula for the first n terms of the given arithmetic sequence.

Key Concepts

Sum Formula

The sum of an arithmetic sequence can be calculated using a specific handy formula. This formula helps find the sum of the first 'n' terms efficiently. This is crucial, especially when dealing with long sequences.

• Firstly, remember that an arithmetic sequence is a series of numbers in which each term after the first is obtained by adding a constant, called the common difference.
• The sum formula for an arithmetic sequence is essentially combining multiple related terms into one calculation.

To compute the sum of an arithmetic sequence, use the formula: \[ S_n = \frac{n}{2} (a_1 + a_n) \]Here:

• \(S_n\) is the sum of the first \(n\) terms.
• \(a_1\) is the first term of the sequence.
• \(a_n\) is the nth term, which can be found from the arithmetic sequence formula.
• \(n\) is the number of terms to be summed.

By using these components, you can easily sum any arithmetic sequence without having to add all the terms individually.

Common Difference

The common difference in an arithmetic sequence is a fundamental concept that helps define the sequence itself. It allows us to move from one term to the next effortlessly.

• A sequence is identified as arithmetic because the difference between successive terms is a constant.
• This constant is known as the common difference, denoted as *d*.

For our case, we have a sequence: 25, 22, 19, 16, ...

• To find the common difference, subtract the first term from the second term: \(22 - 25 = -3\).
• Doing so with successive terms confirms the common difference remains constant throughout, \(-3\).

This negative common difference will decrease the value of each subsequent term by 3, illustrating a falling sequence. Understanding "d" is crucial, as it assists in determining the nth term and thus aids in finding the sequence's sum.

Arithmetic Sequence Formula

The arithmetic sequence formula is the blueprint to finding any term within the sequence, known as the nth term. This formula is vital for understanding the entire sequence and plays a key role in using the sum formula.

• It is denoted as: \(a_n = a_1 + (n-1)d\)
• This showcases a straightforward method to pinpoint any term in the series by knowing the first term and the common difference.
• This formula comes in handy not only for finding individual terms but also for integrating into other calculations like the sum of the series.

Given our sequence:

• The first term \(a_1\) is 25.
• The common difference \(d\) is -3.

Substituting into the formula, the nth term becomes:\[ a_n = 25 + (n-1)(-3) \]This expression allows you to find the value of any term in the sequence, forming the backbone for calculating the total sum of the sequence.