SUMMATION FORMULA - Practice Questions & MCQ

Summation by Sigma( $\mathrm{\Sigma }$ ) Operator
The summation of each term of a sequence or a series can be represented in a compact form, called summation or sigma notation. This summation is represented by the Greek capital letter, Sigma ( $\mathrm{\Sigma }$ ).

For example,
$\mathrm{n}=10$
$\sum _{n=1}^{n}n$, it means the sum of $n$ terms when $n$ varies from 1 to 10

$\sum _{n=1}^{n=10}n=1+2+3+4+5+6+7+8+9+10$
Properties of Sigma Notation
1. $\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}={\mathrm{T}}_1+{\mathrm{T}}_2+{\mathrm{T}}_3+\dots \dots +{\mathrm{T}}_{\mathrm{n}}$, where, ${\mathrm{T}}_{\mathrm{r}}$ is the general term of the series.
2. $\sum _{\mathrm{r}=1}^{\mathrm{n}}({\mathrm{T}}_{\mathrm{r}}\pm {\mathrm{T}}_{\mathrm{r}}^{\prime })=\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}\pm \sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}^{\prime }$ (sigma operator is distributive over addition and subtraction)
3. $\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}{\mathrm{T}}_{\mathrm{r}}^{\prime }\ne \left(\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}\right)\left(\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}^{\prime }\right)$ (sigma operator is not distributive over multiplication)
4. $\sum _{\mathrm{r}=1}^{\mathrm{n}}\frac{{\mathrm{T}}_{\mathrm{r}}}{{\mathrm{T}}_{\mathrm{r}}^{\prime }}\ne \frac{\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}}{\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}^{\prime }}$ (sigma operator is not distributive over division)
5. $\sum _{\mathrm{r}=1}^{\mathrm{n}}\mathrm{a}{\mathrm{T}}_{\mathrm{r}}=\mathrm{a}\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}$ ( a is constant)
6. $\sum _{j=1}^n\sum _{i=1}^n{T}_i{T}_j=\left(\sum _{i=1}^n{T}_i\right)\left(\sum _{j=1}^n{T}_j\right)$ (here i and j are independent)