Problem 51 - Precalculus: Mathematics for Calculus

Question

Find the partial \(\operatorname{sum} S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a=3, \quad d=5, \quad n=20$$

Step-by-Step Solution

Answer

The partial sum \( S_{20} \) is 1010.

Step 1: Identify the General Formula for the Sum

The formula for the partial sum 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 2: Substitute Known Values into the Formula

Using the given values from the problem, substitute \( a = 3 \), \( d = 5 \), and \( n = 20 \) into the formula. \[ S_{20} = \frac{20}{2} (2 \times 3 + (20-1) \times 5) \]

Step 3: Simplify Inside the Brackets

First, calculate the expression inside the brackets. Start with multiplication and then addition:\( 2 \times 3 = 6 \) and \((20-1) \times 5 = 19 \times 5 = 95 \). Now add these results together: \( 6 + 95 = 101 \).

Step 4: Calculate the Partial Sum

Substitute the simplified result back into the equation for the sum:\[ S_{20} = \frac{20}{2} \times 101 \] Divide \( 20 \) by \( 2 \) to get \( 10 \), and then multiply by \( 101 \): \( 10 \times 101 = 1010 \).

Key Concepts

Partial Sum

In the context of arithmetic sequences, the term **partial sum** refers to the sum of a specific number of terms in a sequence. An arithmetic sequence is a series of numbers in which the difference between consecutive terms stays constant. The partial sum is particularly useful when you need to determine the total of only a portion of the sequence, starting from the first term.

• The partial sum is denoted by the symbol \( S_n \), where \( n \) represents the number of terms included in the sum.
• The formula for the partial sum of an arithmetic sequence is \( S_n = \frac{n}{2} (2a + (n-1)d) \).

This formula provides a straightforward method for calculating the sum of several terms without having to individually add each one. First, it divides the number of terms \( n \) by 2, multiplying the result by the expression \( 2a + (n-1)d \), where \( a \) is the first term and \( d \) is the common difference. This allows for the efficient computation of sums, even for a large number of terms.

Common Difference

The **common difference** is a fundamental concept in an arithmetic sequence. It is the consistent difference between any two successive terms in the sequence. This difference remains fixed throughout the sequence, making it predictable and easy to analyze.

• In a formal expression, the common difference is denoted by \( d \).
• The relationship between terms is defined by simply adding the common difference to the preceding term to arrive at the next term.
• For example, in the sequence 3, 8, 13, 18, the common difference \( d = 5 \).

Understanding the consistent difference \( d \) is crucial when working with arithmetic sequences and sums because it helps determine the progression of the sequence, and influences calculations such as the partial sum.

First Term

The **first term** of an arithmetic sequence sets the starting point of the sequence. Known as \( a \), this initial term is the anchor around which the rest of the sequence is structured. Every subsequent term is calculated by adding the common difference \( d \) to this first term.

• Represented by \( a \), the first term is critical in defining the sequence.
• In an expression like \( a, a+d, a+2d, ... \), it is the first item \( a \).
• For the sequence where \( a = 3 \), it means the sequence starts with 3.

The first term is particularly significant when calculating the partial sum, as it is part of the formula \( S_n = \frac{n}{2} (2a + (n-1)d) \). By establishing the initial value, it sets the precedent for all following calculations in the sequence.

Number of Terms

The **number of terms** in an arithmetic sequence, denoted by \( n \), represents how many terms from the start of the sequence will be considered, especially in calculations like the partial sum.

• The number of terms \( n \) influences the length of the sequence being analyzed or summed.
• In the case of partial sums, \( n \) tells us until which term in the sequence we should sum.
• For example, if \( n = 20 \), it implies the sum involves the first 20 terms of the sequence.

Knowing the number of terms \( n \) helps us utilize the partial sum formula efficiently, guiding us on how far into the sequence to consider. This makes it an essential component when handling arithmetic sequences.