Problem 16 - 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). \(\sum_{n=1}^{10}\left(3 a_{n}+2 b_{n}\right)\)

Step-by-Step Solution

Answer

The result is 220.

Step 1: Understanding the given sums

We know that \(\sum_{i=1}^{10} a_{i} = 40\) and \(\sum_{i=1}^{10} b_{i} = 50\). These represent the sums of each sequence \(a_i\) and \(b_i\) from \(i = 1\) to \(10\).

Step 2: Setting up the target expression

We need to calculate the sum \(\sum_{n=1}^{10}(3a_{n}+2b_{n})\). The expression inside the summation, \(3a_n + 2b_n\), is composed of both sequences \(a_n\) and \(b_n\).

Step 3: Applying distribution in the summation

Sum the individual sequences separately: \( \sum_{n=1}^{10}(3a_{n}+2b_{n}) = \sum_{n=1}^{10} 3a_{n} + \sum_{n=1}^{10} 2b_{n} \). This is leveraging the property of summation that allows you to distribute sum terms individually.

Step 4: Factor constants out of the summations

Factor out the constants 3 and 2 from their respective sums: \( 3\sum_{n=1}^{10} a_{n} + 2\sum_{n=1}^{10} b_{n} \). This is possible because the sum of a constant times a sequence is equal to the constant times the sum of the sequence.

Step 5: Substitute the given values into the expression

Substitute the values \(\sum_{n=1}^{10} a_{n} = 40\) and \(\sum_{n=1}^{10} b_{n} = 50\) into the equation: \(3 \times 40 + 2 \times 50\).

Step 6: Calculate the final result

Perform the multiplication and addition: \(3 \times 40 = 120\) and \(2 \times 50 = 100\). Add the results: \(120 + 100 = 220\).

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each term after the first is derived by adding a constant difference to the previous term. This constant is referred to as the "common difference." For example, the sequence 2, 5, 8, 11 is an arithmetic sequence with a common difference of 3.

• Every term in the sequence can be determined using the formula for the \(n\)-th term: \(a_n = a_1 + (n-1) \cdot d\), where \(a_1\) is the first term and \(d\) is the common difference.
• Arithmetic sequences are straightforward to manipulate, especially when employed in summation processes, allowing easy calculations of sum or shift.

Understanding arithmetic sequences is crucial since they frequently appear in various mathematical problems, including problems involving summation.

Summation Properties

Summation is a mathematical notation represented by the Greek letter sigma \(\Sigma\), used to denote the addition of a sequence of numbers, typically following a specific rule or formula. Understanding summation properties is key to efficiently solving problems that require adding series of terms.

• Linearity: The summation operator allows for the addition of sequences independently, such that \( \sum (a_i + b_i) = \sum a_i + \sum b_i \). This is a pivotal property used to separate sums when sequences are combined.
• Constant Factor: In summation, a constant multiple can be factored out, allowing \( \sum c \cdot a_i = c \cdot \sum a_i \), where \(c\) is constant. This is useful when dealing with sequences that involve constant multipliers.

These properties streamline calculations and simplify complex multi-term summations, making them manageable by breaking them down into simpler sums.

Distribution in Summation

Distribution in summation involves the application of distributive properties to enhance the calculation of sums. Our original exercise illustrates how to distribute when dealing with calculated constants and repeated operations.

• This property allows us to break down the expression \(\sum_{n=1}^{10}(3a_{n}+2b_{n})\) into more manageable parts, leading to separate sums for each sequence.
• By leveraging distributive and linear properties, you simplify sums into parts, resulting in calculations like \(3 \sum a_n + 2 \sum b_n\), showcasing how constants remain outside the summation, independent of the sequences.

By understanding how to efficiently distribute and separate terms within sums, you can quickly manage complex summations, reducing them into elementary parts while preserving mathematical correctness.