Problem 71 - Precalculus with Limits

Question

In Exercises 67 - 74, find the partial sum. \( \sum_{n=11}^{30}n - \sum_{n=1}^{10}n \)

Step-by-Step Solution

Answer

The partial sum is 355.

Step 1: Compute the sum of first sequence

First calculate the sum of numbers from 11 to 30 using the formula \( \frac{n*(n+1)}{2} \). Substituting in the values, \( \frac{30*(30+1)}{2} = 465 \). However, in this sequence, we start from the 11th term, not from 1. So, we need to subtract the sum of initial 10 terms from this value.

Step 2:Compute the sum of second sequence

Next, calculate the sum of numbers from 1 to 10 again using the formula \( \frac{n*(n+1)}{2} \). Substituting in the values, \( \frac{10*(10+1)}{2} = 55 \).

Step 3: Subtract second sequence sum from first

Now, subtract the sum of the second sequence (55) from the first (465-55) to find the required partial sum. Therefore, \( 465 - 55 - 55 = 355 \).

Step 4: Final step: Simplify

The computed sum 355 is the final answer.

Key Concepts

Arithmetic sequence

An **arithmetic sequence** is a list of numbers with a definite pattern: the difference between consecutive terms is constant. This constant difference is known as the "common difference." Arithmetic sequences are everywhere: bank savings, calendar dates, and even in nature.

• Example: In the sequence 2, 4, 6, 8, ..., the common difference is 2.
• The formula for the n-th term of an arithmetic sequence is: \( a_n = a + (n - 1) \cdot d \), where \( a \) is the first term and \( d \) is the common difference.

Recognizing an arithmetic sequence can help simplify mathematical problems. By identifying the first term and the common difference, you can find any term in the sequence quickly. This comes in handy when determining sums of parts of sequences.

Summation formula

In mathematics, the **summation formula** provides a quick way to add up a list of numbers, allowing you to compute the sum of terms in a sequence efficiently. This is particularly useful for arithmetic sequences.To find the sum of an arithmetic sequence, the formula is:\[ S_n = \frac{n}{2} \cdot (a + l) \]where:

• \( S_n \) is the total sum of the first \( n \) terms
• \( n \) is the total number of terms
• \( a \) is the first term
• \( l \) is the last term.

Summation not only simplifies the process of adding a large number of terms but also saves a significant amount of time. Instead of adding numbers one by one, this formula allows you to obtain the sum swiftly, which is why it was used in the original exercise to add numbers from 11 to 30 and from 1 to 10.

Series

A **series** is the expression formed by adding the terms of a sequence. When we talk about a series, we often mean summing elements of a sequence over a defined range. Types of series include:

• **Finite series**: A series with a limited number of terms, like the sum of numbers from 1 to 10.
• **Infinite series**: Continues indefinitely, like the sum of all terms in a sequence without an ending point.

When dealing with finite arithmetic series, the goal is often to find the "partial sum," which is the sum of a portion of the entire sequence. In problems like the original exercise, finding partial sums helps understand patterns and totals within specific parts of the series. This skill is valuable across various fields such as finance, computer science, and engineering, where efficient calculations are crucial.