Algebra

Evaluate the following sums.

1. $\displaystyle\sum_{i=3}^{5}(7ix+11)$

2. $\displaystyle\sum_{x=3}^{5}(7ix+11)$

3. $\displaystyle\sum_{i=-3}^{-5}(-7ix-11)$

4. $\displaystyle\sum_{i=4}^{8}ijk$

5. $\displaystyle\sum_{j=4}^{8}ijk$

6. $\displaystyle\sum_{n=4}^{8}ijk$

7. $\displaystyle\sum_{i=3}^{7}13$

8. $\displaystyle\sum_{i=1}^{3}i^{2}+\sum_{j=4}^{6}j^{2}$

9. $\displaystyle\sum_{i=1}^{3}\left(\sum_{j=5}^{7}(i+j)\right)$

Given $\displaystyle\sum_{i=1}^{5}a_{i}^{2}=100$, $\displaystyle\sum_{i=2}^{5}a_{i-1}=10$, and $\displaystyle\sum_{i=1}^{5}(a_{i}+2)^{2}=200$, what is $a_{5}$?

Solve the inequality $x^{2}+15<8|x|$.

Prove 1.36.

Find the coefficient of $x^{7}$ in $(7x-3)^{9}$.

Suppose that commutativity of multiplication (1.7) is no longer true. Expand $(x+y)^{3}$.

By using the substitution $u=x^{2}-2x-2$, find all $x\in\mathbb{R}$ such that

Let $x\in[0,3]$.

1. Expand $(x-2)(x^{2}-3x+1)$.

2. Hence find the range of values of $x$ such that

Solve the inequality

given that $x\geq 0$.

Let $N$ be a positive integer.

1. Express $\frac{7x+4}{x^{3}+3x^{2}+2x}$ in the form $\frac{A}{x}+\frac{B}{x+1}+\frac{C}{x+2}$, where $A$, $B$, and $C$ are real constants to be determined.

2. Hence simplify

Prove that

for all non-negative integers $n$.

Let $x_{1},x_{2},\dots,x_{n}\in\mathbb{R}$.

1. Prove that $|xy|=|x||y|$ for all $x,y\in\mathbb{R}$.

2. Prove that $|x_{1}x_{2}\cdots x_{n}|=|x_{1}||x_{2}|\cdots|x_{n}|$.