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 a1 and last term an:
Given the first term a1 and the common ratio r
The product of the first n terms of a Geometric Progression is given by the following:
Given the first term a1 and last term an:
Given the first term a1 and the common ratio r
Symbols and Notations
γ, γm = Unit weight, bulk unit weight, moist unit weight
γd = Dry unit weight
γsat = Saturated unit weight
γb, γ' = Buoyant unit weight or effective unit weight
γs = Unit weight of solids
γw = Unit weight of water (equal to 9810 N/m3)
W = Total weight of soil
Ws = Weight of solid particles
Ww = Weight of water
V = Volume of soil
Vs = Volume of solid particles
Vv = Volume of voids
Vw = Volume of water
S = Degree of saturation
w = Water content or moisture content
G = Specific gravity of solid particles
The length of arc in rectangular coordinates is given by the following formulas:
See the derivations here: http://www.mathalino.com/reviewer/integral-calculus/length-arc-xy-plane…
The three-moment equation gives us the relation between the moments between any three points in a beam and their relative vertical distances or deviations. This method is widely used in finding the reactions in a continuous beam.
Consider three points on the beam loaded as shown.
Quadrilateral circumscribing a circle (also called tangential quadrilateral) is a quadrangle whose sides are tangent to a circle inside it.
Area,
Where r = radius of inscribed circle and s = semi-perimeter = (a + b + c + d)/2
Derivation for area
For a triangle of given three sides, say a, b, and c, the formula for the area is given by
where s is the semi perimeter equal to P/2 = (a + b + c)/2.
Ptolemy's theorem for cyclic quadrilateral states that the product of the diagonals is equal to the sum of the products of opposite sides. From the figure below, Ptolemy's theorem can be written as
For a cyclic quadrilateral with given sides a, b, c, and d, the formula for the area is given by
Where s = (a + b + c + d)/2 known as the semi-perimeter.
The lateral area of frustum of a right circular cone is given by the formula
where
R = radius of the lower base
r = radius of the upper base
L = length of lateral side