Problem 45 - Precalculus Mathematics for Calculus

Question

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 Parameters

We need to find the partial sum \( S_n \) of an arithmetic sequence, where \( a = 4 \) (the first term), \( d = 2 \) (the common difference), and \( n = 20 \) (the number of terms).

Step 2: Use the Formula for Partial Sum

The formula for the partial sum \( S_n \) of an arithmetic sequence is \( S_n = \frac{n}{2} (2a + (n-1)d) \).

Step 3: Substitute the Values into the Formula

Substitute \( a = 4 \), \( d = 2 \), and \( n = 20 \) into the formula: \[ S_{20} = \frac{20}{2} (2 \times 4 + (20-1) \times 2) \].

Step 4: Simplify the Expression

First, simplify the terms inside the parentheses: - \( 2 \times 4 = 8 \),- \( 20 - 1 = 19 \),- \( 19 \times 2 = 38 \).Now substitute back: \[ S_{20} = 10 (8 + 38) \].

Step 5: Calculate the Final Sum

Add the results inside the parentheses:- \( 8 + 38 = 46 \).Now calculate the product:\[ S_{20} = 10 \times 46 = 460 \].

Key Concepts

Partial Sum

A partial sum represents the sum of a specific number of initial terms in a sequence. For arithmetic sequences, this sum is crucial when you need to find how much the sequence adds up to over a certain number of terms. The formula to compute this sum, given the number of terms, the first term, and the common difference, is \( S_n = \frac{n}{2} (2a + (n-1)d) \).
This formula makes it easy because:

• It accounts for each term in the sequence.
• It allows you to quickly calculate the total sum without finding each individual term first.

Understanding how to use this formula effectively can greatly aid in solving problems involving arithmetic sequences.

First Term

The first term of an arithmetic sequence is the starting value from which the sequence is built. In any arithmetic sequence, it is represented by the letter \( a \). Knowing the first term is essential as it sets the base for generating the rest of the sequence.
For instance, in the sequence where \( a = 4 \), this value indicates that the sequence begins with 4. From this starting point, each subsequent term is reached by adding the common difference to the previous term. The first term, therefore, acts as the entry point for diving deeper into the patterns within the sequence.

Common Difference

The common difference in an arithmetic sequence is the consistent interval between consecutive terms. This is denoted by the letter \( d \). It tells you how much each term increases or decreases as you move from one term to the next.
For example, if \( d = 2 \), each term is 2 units larger than the one before it. This steady increase is what makes arithmetic sequences predictable and simple to work with. Identifying this common difference allows you to understand the pattern of expansion or contraction within the sequence and ensures correct calculation of further terms or partial sums.

Number of Terms

The number of terms in an arithmetic sequence, represented by \( n \), determines how many terms you sum up or consider within the sequence. This value is crucial when calculating a partial sum because it dictates the extent or length of your calculation.
In our example, \( n = 20 \) indicates that we're interested in the first 20 terms of the sequence. Knowing the number of terms is vital for using the partial sum formula, as it ensures you neither over-count nor under-count the terms in your sum.