Problem 17 - Calculus

Question

Suppose that \(\sum_{i=1}^{10} a_{i}=40\) and \(\sum_{i=1}^{10} b_{i}=50 .\) Calculate each of the following (see Example 1). \(\sum_{p=0}^{9}\left(a_{p+1}-b_{p+1}\right)\)

Step-by-Step Solution

Answer

The sum is \(-10\).

Step 1: Define the Given Series

We are given that \( \sum_{i=1}^{10} a_i = 40 \) and \( \sum_{i=1}^{10} b_i = 50 \). Our goal is to find the sum \( \sum_{p=0}^{9}(a_{p+1} - b_{p+1}) \). Notice that the indices of the series we need to calculate are shifted compared to the given sums.

Step 2: Shift the Indices

The sum \( \sum_{p=0}^{9}(a_{p+1} - b_{p+1}) \) involves terms from \( a_1 \) to \( a_{10} \) and from \( b_1 \) to \( b_{10} \). These are exactly the terms involved in the given sums \( \sum_{i=1}^{10} a_i \) and \( \sum_{i=1}^{10} b_i \).

Step 3: Break Down the Desired Sum

Break the expression down to separate the sums: \( \sum_{p=0}^{9}(a_{p+1} - b_{p+1}) = \sum_{p=0}^{9} a_{p+1} - \sum_{p=0}^{9} b_{p+1} \). Because summing from index 0 to 9 covers all indices 1 to 10, we can rewrite this as: \( \sum_{i=1}^{10} a_i - \sum_{i=1}^{10} b_i \).

Step 4: Substitute the Known Values

Substitute the known sums into the expression: \( \sum_{i=1}^{10} a_i - \sum_{i=1}^{10} b_i = 40 - 50 \).

Step 5: Calculate the Result

Perform the subtraction: \( 40 - 50 = -10 \). This gives us the value of the required sum.

Key Concepts

Series and Sums

Series and sums are fundamental concepts in calculus and mathematics. When we talk about a series, we refer to the sum of a sequence of numbers. For example, if we have a sequence of numbers like \(a_1, a_2, \ldots, a_n\), the series is noted as \(\sum_{i=1}^{n} a_i\), which means we sum all terms from the first to the \(n\)-th term.

• The symbol \(\sum\) (sigma) signifies summation, which is the process of adding together a succession of terms.
• The lower part of the sigma notation indicates the starting index, while the upper part denotes the end index.

Summations are often used to simplify expressions and solve problems that involve adding multiple numbers together. In calculus, this concept extends to infinite series, where the numbers continue indefinitely, and we explore their convergence or divergence.
In the given problem, we calculated the difference between two series, \(\sum_{i=1}^{10} (a_i)\) and \(\sum_{i=1}^{10} (b_i)\), by rearranging the terms to match the given sums.

Mathematical Indices

The concept of indices in mathematics refers to the positions of elements within a series or summation. Indices make it possible to refer to different elements in a set in a convenient and organized manner. In the exercise, we deal with shifted indices, as we need to compute \(\sum_{p=0}^{9} (a_{p+1} - b_{p+1})\).

• The indices \(i\) and \(p+1\) indicate which term in the sequence we are referring to.
• Adjusting indices is essential when translating one series to match another's given form, particularly when elements have been shifted or offset in the series.

The shift from \(i=1\) to \(10\) and \(p=0\) to \(9\) implies the same terms, but they are just expressed differently due to their starting points. This small index shift doesn’t alter the elements in the series; instead, it explicates how the terms are organized. Comprehending indices aids in understanding and solving complex problems involving sequences and series accurately.

Basic Arithmetic Operations

Arithmetic operations, such as addition and subtraction, are crucial in day-to-day calculations and problem-solving in mathematics. In the context of series and calculations like the one in the problem, these operations are tools to simplify and solve expressions.

• Addition involves combining the values of two or more numbers to get a total. It is symbolized by \(+\).
• Subtraction, indicated by \(-\), is about finding the difference between numbers, which is the operation we relied on to solve the given series problem.

In the given solution, arithmetic helped us evaluate \(40 - 50\), leading to the answer \(-10\). Recognizing how each operation functions within mathematical expressions is key to correctly interpreting and solving problems like calculating sums of series and their differences. The practice of breaking down equations step-by-step not only clarifies the solution but also reinforces the understanding of how basic operations interact.