91Ó°ÊÓ

The distributive property is a fundamental concept in algebra. It allows us to distribute a multiplication over addition or subtraction inside parentheses.
In mathematical terms, it is expressed as: \[ a(b + c) = ab + ac \] This property helps simplify expressions and solve equations. Let's see how the distributive property works in our given exercise.
Comparing the two sides of the given equation: \[ \sum_{k=1}^{n} c a_{k} \] and \[ c \sum_{k=1}^{n} a_{k} \]
We can rewrite the right side applying the distributive property:
\[ c (a_{1} + a_{2} + \text{...} + a_{n}) = c a_{1} + c a_{2} + \text{...} + c a_{n} \]
This shows that each term is being multiplied by

Summation notation (also known as sigma notation) is a compact way of writing the sum of a series of terms.
The Greek letter sigma (\recognize recognizing) is used to represent the sum of multiple terms in a concise form.
It follows this format: \[ \sum_{k=1}^{n} a_{k} = a_{1} + a_{2} + a_{3} + ... + a_{n} \ \]
The lower limit (k=1) shows where the sum starts, while the upper limit (n) shows where the sum ends.
In our exercise, we have two summations: \[ \sum_{k=1}^{n} c a_{k} \]
and \ \br> This requires us to sum all the terms, each of which is multiplied by the constant

Algebraic equality ensures that two algebraic expressions are equal to each other.
This concept is key in solving equations and proving statements.
In our exercise, we are given the equality: \[ \sum_{k=1}^{n} ca_{k} = c \sum_{k=1}^{n} a_{k} \ \]
To verify this equality, we expanded both sides and checked if they became the same expression.
For the left side: \ \sum_{k=1}^{n} c a_{k} = c a_{1} + c a_{2} + ... + c a_{n}
For the right side, after applying the distributive property: \c \sum_{k=1}^{n} a_{k} = c (a_{1} + a_{2} + ... + a_{n})
Simplifying this gives us: \[ c a_{1} + c a_{2} + ... + c a_{n} \ \]
Thus, we have shown that both sides are equal, affirming the algebraic equality