Problem 43 - Algebra and Trigonometry

Question

\(43-48\) . 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: Understand the Problem

We need to find the partial sum \( S_n \) of an arithmetic sequence with first term \( a = 1 \), common difference \( d = 2 \), and number of terms \( n = 10 \).

Step 2: Recall the Formula for Partial Sum of an Arithmetic Sequence

The formula to find the sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is given by: \[ S_n = \frac{n}{2} (2a + (n-1) d) \] where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms.

Step 3: Substitute Given Values into the Formula

We substitute \( a = 1 \), \( d = 2 \), and \( n = 10 \) into the sum formula: \[ S_{10} = \frac{10}{2} [2 \times 1 + (10-1) \times 2] \]

Step 4: Calculate Inside the Brackets

First, calculate \( 2 \times 1 + (10-1) \times 2 \): 1. \( 2 \times 1 = 2 \)2. \( 10-1 = 9 \)3. \( 9 \times 2 = 18 \)4. Thus, \( 2 + 18 = 20 \).

Step 5: Calculate the Sum

Now, plug this back into the outside part of the formula: \[ S_{10} = \frac{10}{2} \times 20 \] Calculate: \( \frac{10}{2} = 5 \) and \( 5 \times 20 = 100 \).

Step 6: Verify the Result

Double-check calculations to ensure no errors were made in the calculations for the expression and the final sum.

Key Concepts

Partial Sum

In an arithmetic sequence, the partial sum represents the sum of a certain number of terms at the beginning of the sequence. This is particularly useful when you want to find out the total value for a set amount of terms without having to add each one individually.
To find the partial sum of an arithmetic sequence, we use the formula:

• \( S_n = \frac{n}{2} (2a + (n-1) d) \)

This formula allows you to quickly calculate the sum of the first \( n \) terms in the sequence. Just plug in the values for the first term, common difference, and number of terms to get your answer.
In our example, we're finding the sum of the first 10 terms, which is calculated as 100. This figure represents the aggregate value of the initial 10 numbers in the progression.

Common Difference

The common difference in an arithmetic sequence is a consistent value that separates each term from the next. Imagine stepping stones that are evenly spaced; the common difference is the distance between each stone.
This value is crucial because it defines the sequence's regular spacing and progression.

• For instance, in our exercise, every next term is 2 more than the previous one, making the common difference \( d = 2 \).

The common difference helps in building the sequence from the first term, ensuring each new term increases or decreases uniformly. This is an essential concept in forming the basis of arithmetic sequences.

First Term

The first term of an arithmetic sequence is simply the initial number from which the entire sequence starts. It sets the foundation of the sequence.
By knowing the first term, you have a starting point to calculate any subsequent term by applying the common difference repeatedly.

• In the discussed exercise, the first term is \( a = 1 \).

This means our sequence begins at 1, and all subsequent terms are calculated based on this value. The starting term is essential because, without it, you cannot accurately determine the remaining terms in the sequence.

Number of Terms

The number of terms in an arithmetic sequence indicates how many elements are included, starting from the first term. This tells us how extensive the sequence is when calculating things like partial sums.

• In the problem you are examining, the given number of terms is \( n = 10 \).

Knowing the number of terms helps you apply formulas, like the partial sum formula, correctly. It ensures that when calculating totals, every intended term is considered.
Having this count helps you determine the endpoints of your calculations, ensuring that you sum or list the exact number of desired terms, no more and no less.