Problem 39 - Precalculus: Mathematical for Calculus

Question

Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a=1, d=2, n=10$$

Step-by-Step Solution

Answer

The partial sum \(S_{10}\) is 100.

Step 1: Identify the General Formula for the Arithmetic Sequence

An arithmetic sequence is defined by the first term, \(a\), and the common difference, \(d\). The general formula for the \(n^{th}\) term of an arithmetic sequence is \(a_n = a + (n-1)\cdot d\).

Step 2: Calculate the Last Term of the Sequence

Substitute the known values into the formula for the \(n^{th}\) term: \(a_n = 1 + (10-1)\cdot 2 = 1 + 18 = 19\). Thus, the 10th term, \(a_{10}\), is 19.

Step 3: Apply the Formula for the Partial Sum

The partial sum \(S_n\) of the first \(n\) terms of an arithmetic sequence can be calculated using the formula \(S_n = \frac{n}{2} \times (a + a_n)\).

Step 4: Substitute Values into the Partial Sum Formula

Substitute \(n = 10\), \(a = 1\), and \(a_{10} = 19\) into the partial sum formula: \(S_{10} = \frac{10}{2} \times (1 + 19) = 5 \times 20 = 100\).

Step 5: Conclusion of the Partial Sum Calculation

The partial sum of the first 10 terms of the arithmetic sequence is \(S_{10} = 100\).

Key Concepts

Partial Sum

A partial sum in an arithmetic sequence refers to the sum of a certain number of terms in that sequence. In the given exercise, the partial sum is denoted as \( S_n \), where \( n \) represents the number of terms we want to sum up.
To find the partial sum of an arithmetic sequence, we can use the formula:

\( S_n = \frac{n}{2} \times (a + a_n) \)

This formula is quite handy as it allows us to sum a specific number of terms without having to add each term one by one. In the context of our example, where \( n = 10 \), the partial sum formula aggregates the first 10 terms of the sequence, resulting in a total of 100.
Understanding how to apply the partial sum formula is crucial for efficiently dealing with large sequences and simplifying calculations.

Common Difference

The common difference in an arithmetic sequence is what makes this kind of sequence special. It is the amount each term increases (or decreases) from the previous one. In simple terms, it is the difference that remains constant between consecutive terms.
Using the given values, the common difference \( d = 2 \), we know that each term in the sequence is 2 units more than the last. For instance:

• The first term is 1.
• The second term is 1 + 2 = 3.
• The third term is 3 + 2 = 5, and so on.

Knowing the common difference helps in predicting subsequent terms and calculating the general formula without having to manually compute each term. It's one of the foundational concepts for dealing with arithmetic sequences.

Nth Term

Finding the nth term of an arithmetic sequence is key to understanding its structure. The nth term tells us the value of the sequence at a specific position. The formula for finding the nth term \( a_n \) is:

\( a_n = a + (n-1) \cdot d \)

Where:

• \( a \) is the first term of the sequence,
• \( n \) is the term position,
• \( d \) is the common difference.

In the exercise, to calculate the 10th term, we substitute the values: \( a = 1 \), \( n = 10 \), and \( d = 2 \). This yields:

\( a_{10} = 1 + (10-1) \cdot 2 = 19 \)

This step helps us identify individual terms quickly and accurately. Understanding how to determine the nth term is essential for tasks such as finding a specific term's value or checking if a certain number belongs to the sequence.

General Formula

The general formula of an arithmetic sequence describes how any term \( a_n \) can be computed based on its position \( n \). This formula universally applies to all arithmetic sequences and is given by:

\( a_n = a + (n-1) \cdot d \)

By using the first term \( a \) and the common difference \( d \), you can find any term in the sequence. It's useful because it combines the sequence's starting point and growth rate into one concise expression.
For our exercise example with \( a = 1 \) and \( d = 2 \), the general formula allows us to quickly find any term by simply plugging in the value of \( n \). Understanding how to effectively use this formula makes working with arithmetic sequences straightforward, enabling you to approach complex sequence problems with confidence.