Problem 7 - Calculus Volume 1

Question

Suppose that \(\sum_{i=1}^{100} a_{i}=15\) and \(\sum_{i=1}^{100} b_{i}=-12 .\) In the following exercises, compute the sums. $$\sum_{i=1}^{100}\left(5 a_{i}+4 b_{i}\right)$$

Step-by-Step Solution

Answer

The sum is 27.

Step 1: Use the Properties of Sums

The sum of a linear combination \( \sum_{i=1}^{n} (c a_i + d b_i) = c \sum_{i=1}^{n} a_i + d \sum_{i=1}^{n} b_i \). This means we can separate the terms in the sum and multiply each sum by its respective coefficient.

Step 2: Insert Known Sums into the Formula

Substitute \( c = 5 \) and \( d = 4 \) into the formula from Step 1. We have:\[ \sum_{i=1}^{100}(5 a_i + 4 b_i) = 5 \sum_{i=1}^{100} a_i + 4 \sum_{i=1}^{100} b_i. \]Insert the known sums:\( \sum_{i=1}^{100} a_i = 15 \) and \( \sum_{i=1}^{100} b_i = -12 \).

Step 3: Calculate Each Component

Calculate each part:\( 5 \cdot 15 = 75 \)\( 4 \cdot (-12) = -48 \)

Step 4: Find the Final Sum

Add the results of the two calculated components together:\[ 75 + (-48) = 27. \]

Step 5: Conclusion

Therefore, the sum \( \sum_{i=1}^{100}(5 a_i + 4 b_i) \) equals 27.

Key Concepts

Summation Notation

Summation notation is a way to compactly represent the sum of a series of terms. It's written using the Greek letter sigma (\( \Sigma \)), and it details the process of summation over a sequence of terms. Let's break down how it works:

• The expression \( \sum_{i=1}^{n} a_i \) means "add up all of the \( a_i \) terms from \( i = 1 \) to \( i = n \)."
• The letter \( i \) is called the index of summation, which iterates over each specified term.
• The lower and upper limits of the sum, such as 1 and \( n \), tell us where to start and stop adding.

Understanding summation notation is crucial because it allows complex equations to be expressed in a more manageable form. Instead of writing a long addition operation, we write it succinctly using this format. In operations such as the one in the exercise, comprehension of this notation simplifies calculations and computational processes.

Properties of Sums

The properties of sums are rules that help simplify complex summations. A key property is the ability to break apart sums involving linear combinations:

• This rule states: \( \sum_{i=1}^{n} (c a_i + d b_i) = c \sum_{i=1}^{n} a_i + d \sum_{i=1}^{n} b_i \).
• This works by distributing the constants \( c \) and \( d \) over their respective summations.
• Simplifying calculations is easier by breaking the original sum into separate, smaller sums.

By using these properties, we can compute sums by handling each component separately, as shown in the solution steps. This makes it more efficient to tackle complex sums by breaking them down into simpler parts.

Linear Combinations

A linear combination involves creating a new equation by adding and subtracting multiples of given quantities. In calculus and algebra, we often see expressions like \( c a_i + d b_i \), which are linear combinations of terms \( a_i \) and \( b_i \).

• Linear combinations maintain the "linearity" property, meaning the operation is both additive and homogeneous.
• This allows us to manipulate expressions flexibly within limitations set by given coefficients, \( c \) and \( d \).
• For instance, in the exercise, terms are computed individually using each respective linear combination, leveraging their known sum values.

Understanding linear combinations allows you to rearrange and simplify equations elegantly, often leading to easier computation of sums, especially in exercises similar to the one provided.