Problem 27 - Algebra 2 and Trigonometry

Question

Show that \(\sum_{i=1}^{n} k a_{i}=k \sum_{i=1}^{n} a_{i}\)

Step-by-Step Solution

Answer

The equation is true due to the distributive property of multiplication over addition in summations.

Step 1: Understand the Problem Statement

The problem asks us to show a property of summation involving a constant factor. Specifically, we need to prove that multiplying each term in a summation by a constant is equivalent to multiplying the entire summation by that constant.

Step 2: Write the Left-hand Side of the Equation

The left-hand side of the equation is the summation of the terms where each term is multiplied by the constant factor \(k\). It is represented as \(\sum_{i=1}^{n} k a_{i}\).

Step 3: Factor Out the Constant

Since multiplication is distributive over addition, we can factor out the constant \(k\) from the summation. Thus, \(\sum_{i=1}^{n} k a_{i} = k \sum_{i=1}^{n} a_{i}\).

Step 4: Write the Right-hand Side of the Equation

The right-hand side is already written as \(k \sum_{i=1}^{n} a_{i}\), which matches the expression we derived in Step 3.

Step 5: Conclude the Proof

Since both the manipulated left-hand expression and the right-hand side of the equation are identical, we have shown that the original statement \(\sum_{i=1}^{n} k a_{i} = k \sum_{i=1}^{n} a_{i}\) is true.

Key Concepts

Factorization

Factorization is an essential mathematical concept, especially when dealing with expressions or equations that need simplification. In this context, factorization means breaking down an expression into a product of its factors. A factor is a number or expression that divides another number or expression without leaving a remainder. In the exercise provided, we dealt with the expression \(\sum_{i=1}^{n} k a_{i}\). The process of factorization is utilized to rewrite this expression more simply. By identifying the constant factor \(k\) that multiplies each term of the series, we can "factor out" \(k\) from the summation. This means we recognize that \(k\) is a common factor of all terms in the summation, allowing us to write:- \(\sum_{i=1}^{n} k a_{i} = k \sum_{i=1}^{n} a_{i}\)This step greatly simplifies the evaluation of the expression and is a key factorization technique that can be applied to similar problems across different areas of mathematics. Factorization helps make calculations easier and more manageable by reducing complexity.

Summation Notation

Summation notation is a mathematical symbol used to represent the sum of a sequence of terms. It is concise and allows us to write long sums in a simple format. The symbol \(\sum\) denotes the summation. When combined with limits, it tells us what series of terms we are summing over.In the exercise, the notation \(\sum_{i=1}^{n} a_{i}\) means we add up the terms \(a_1, a_2, ..., a_n\). The variable \(i\) is known as the index of summation, and it takes on each integer value from the lower limit (1 in this case) to the upper limit (\(n\)).Using summation notation makes it straightforward to express large sums succinctly:

• \(\sum_{i=1}^{n} a_{i}\) can expand to \(a_1 + a_2 + a_3 + ... + a_n\)

This allows us to easily manipulate and apply algebraic properties to entire sums, as demonstrated in the exercise.

Distributive Property

The distributive property is a fundamental property of numbers and operations, which states that for any numbers or expressions \(a\), \(b\), and \(c\), the expression \(a(b + c) = ab + ac\) holds. This property applies to multiplication over addition or subtraction, ensuring that multiplication distributed over terms inside a parenthesis.In the exercise, we used the distributive property to simplify the summation \(\sum_{i=1}^{n} k a_{i}\). Each term \(k a_{i}\) can be viewed as \(k\) multiplied by the individual terms \(a_i\). The distributive property allows us to factor \(k\) out of each term in the summation:

• \(\sum_{i=1}^{n} k a_{i} = k \sum_{i=1}^{n} a_{i}\)

This demonstrates that multiplying the entire sum by \(k\) is the same as multiplying each term by \(k\) and then summing. It simplifies the calculation and is a clear example of how distributive property aids in mathematical simplification and manipulation.