Question

How do you write sum in expanded form?

Answer 1

Here we can express sum in expanded form by using summation notation. And also we see how the summation notation will be used.

Complete step-by-step solution:
In this section we need to do a brief review of summation notation or sigma notation. We will start out with two integers $n$ and $m$ with $n < m$ and a list of numbers denoted as follows
${a_n},{a_{n + 1}},........{a_{m - 2}}, {a_{m - 1}},{a_m}$ we want to add them up in other words we want,
${a_n} + {a_{n + 1}} + ........{a_{m - 2}} + {a_{m - 1}}, + {a_m}$ we can denoted this case as follows$\sum\limits_{i = n}^m {{a_i}} = {a_n} + {a_{n + 1}} + ........{a_{m - 2}} + {a_{m - 1}}, + {a_m}$ this notation is called summation notation or sigma notation.
Summation notation:
Sigma notation is used as a convenient shorthand notation for the summation of terms for example, we write $\sum\limits_{n = 1}^5 n = 1 + 2 + 3 + 4 + 5$
Here the symbol $\sum {}$ sigma indicates a sum. The numbers at the top and bottom of sigma are called boundaries and tell us what numbers we substitute into the expression for the terms in our sum. What comes after sigma is an algebraic expression representing terms in the sum. In the example above, $n$ is a variable and represents the terms in our sum.

Note: The symbol that is often used to express the concept of summation is the uppercase Greek letter, sigma $\sum {}$. The $\sum {}$notation is used in the following form $\sum\limits_{i = 1}^n {{u_i}}$ . The notation is read as the summation of all the ${u_i}$’s from $i = 1$ to $i = n$ where $n$ is the number of terms as long as $i = 1$ and ${u_i}$ represents the terms that are being added and $i$ is the variable that is used to increment to the next term.
We started the series at ${i_0}$ (this means that initial value of $i$ ) to denote the fact that they can start at any value of $i$ that we need them to.
Also note that while we can break up sums and differences as we did.
But we can’t do the same thing for products and quotients.