Problem 45 - Algebra and Trigonometry

Question

\(43-48\) . Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$ a=4, d=2, n=20 $$

Step-by-Step Solution

Answer

The partial sum \(S_{20}\) is 460.

Step 1: Identify the First Term

The first term of the arithmetic sequence is given as \(a = 4\). This will serve as the starting point of the sequence.

Step 2: Determine the Common Difference

The common difference \(d\) in the arithmetic sequence is given as \(d = 2\). This means each subsequent term increases by 2.

Step 3: Calculate the Last Term

To find the last term \(a_n\) of the sequence, use the formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a + (n-1) \times d\). Here, substituting the values gives:\[a_{20} = 4 + (20-1) \times 2 = 4 + 38 = 42\].

Step 4: Use the Partial Sum Formula

The sum of the first \(n\) terms \(S_n\) of an arithmetic sequence can be calculated with the formula \(S_n = \frac{n}{2} \times (a + a_n)\). Here, substitute the known values:\[S_{20} = \frac{20}{2} \times (4 + 42) = 10 \times 46 = 460\].

Step 5: Verify the Partial Sum

Double-check the calculation of \(S_{20}\) by ensuring that all substitutions and arithmetic operations are performed correctly. The formula and substitutions yield:\[S_{20} = 10 \times 46 = 460\].

Key Concepts

Partial Sum

A partial sum in an arithmetic sequence represents the sum of a specified number of consecutive terms from the sequence. It is a way to find the accumulation of terms up until a certain term, often denoted as \( S_n \).
Understanding partial sums is essential when you need to determine how much of the sequence is included up to a certain point.
For example, if you have the sequence with a first term of 4 and you want to know the sum of the first 20 terms, the partial sum provides that total sum.
By using the partial sum formula, you can quickly calculate the exact sum without needing to add each term individually.
This is especially useful in long sequences where manual addition would be time-consuming.

Common Difference

In any arithmetic sequence, the term "common difference" (denoted \(d\)) is fundamental. It refers to the constant amount by which each term in the sequence increases or decreases from the previous term.

• If the common difference is positive, each term is larger than the one before it.
• If the common difference is negative, each term is smaller than the one before it.
• A common difference of zero means all terms in the sequence are the same.

In our exercise, the common difference is 2, indicating a consistent increase of 2 for each successive term in the sequence.
Understanding the common difference helps you predict the behavior of the sequence and is key to identifying any arithmetic sequence.

First Term

The first term of an arithmetic sequence is usually denoted \(a\). It serves as the starting point or the anchor of the entire sequence.
The first term is crucial because it sets the initial value from which the sequence progresses.
With a known common difference, you can determine every subsequent term starting from this first term.
For instance, in our exercise, the first term \(a\) is given as 4.
From here, each following term is built by adding the common difference to this initial value.

Sum Formula

The sum formula for an arithmetic sequence is a valuable tool for calculating the sum of a certain number of terms without manually adding each one.
The formula is expressed as:\[ S_n = \frac{n}{2} \times (a + a_n) \] where \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, \( a_n \) is the \( n \)-th term, and \( n \) is the number of terms.
In our problem, plugging in the known values into this formula allowed us to find that the partial sum \( S_{20} \) equals 460.
This formula streamlines the process of summing terms and ensures you accurately obtain the total.
It's particularly beneficial for larger sequences where manual addition isn't practical.