## Derivation of Product of First n Terms of Geometric Progression

The product of the first $n$ terms of a Geometric Progression is given by the following:

Given the first term $a_1$ and last term $a_n$:

Given the first term $a_1$ and the common ratio $r$

## Factor trinomials mentally! Tips and tricks

Here are some tips and tricks in factoring the trinomial $ax^{2}+bx+c$ mentally. Once you master the techniques in this blog, you can simplify expressions and solve equations that require factoring with "lightning" speed, and impress your friends.

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## Problem 5: Evaluate $\displaystyle \int \dfrac{(6z - 1) \, dz}{\sqrt{(2z + 1)^3}}$ by Algebraic Substitution

**Problem**

Evaluate $\displaystyle \int \dfrac{(6z - 1) \, dz}{\sqrt{(2z + 1)^3}}$

## Problem 4: Evaluate $\displaystyle \int \dfrac{y \, dy}{\sqrt[4]{1 + 2y}}$ by Algebraic Substitution

**Problem**

Evaluate $\displaystyle \int \dfrac{y \, dy}{\sqrt[4]{1 + 2y}}$

**Answer Key**

## Problem 3: Evaluate $\displaystyle \int \dfrac{x^3 \, dx}{(x^2 + 1)^3}$ by Algebraic Substitution

**Problem**

Evaluate $\displaystyle \int \dfrac{x^3 \, dx}{(x^2 + 1)^3}$

## Problem 2: Evaluate $\displaystyle \int y^3\sqrt{2y^2 + 1} \,\, dy$ by Algebraic Substitution

**Problem**

Evaluate $\displaystyle \int y^3\sqrt{2y^2 + 1} \,\, dy$

## Which curve has a constant first derivative?

**Problem**

If the first derivative of the equation of a curve is a constant, the curve is:

A. circle | C. hyperbola |

B. straight line | D. parabola |

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## Depth and vertex angle of triangular channel for minimum perimeter

**Problem**

A triangular shaped channel is to be designed to carry 700 L/s on a slope of 0.0001. Determine what vertex angle and depth of water over the vertex will be necessary to give a section with minimum perimeter, assuming the channel is made of timber, *n* = 0.012. Use Manning’s formula.

A. θ = 45°, h = 1.425 m |
C. θ = 45°, h = 2.125 m |

B. θ = 90°, h = 2.215 m |
D. θ = 90°, h = 1.215 m |