Given the geometric sequence where a1 = 3 and the common ratio is −1, what is the domain for n?
2 answers:
Answer:
The domain for n is:
All integers where n ≥ 1
Step-by-step explanation:
We are given a geometric sequence where the first term of the sequence is:
and the common ratio(r) is −1.
We are asked to find the domain for n.
We know that the domain of the geometric sequence whose terms are given by are all the natural numbers since by putting the n we get the corresponding term.
Hence, the domain is:
All integers where n ≥ 1
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Answer:
√3 is irrational
Step-by-step explanation:
The location of the third point of a triangle can be found using a rotation matrix to transform the coordinates of the given points.
<h3 /><h3>Location of point C</h3>With reference to the attached figure, the slope of line AC is √3, an irrational number. This means the line AC <em>never passes through a point with integer coordinates</em>. (Any point with integer coordinates would be on a line with rational slope.)
<h3>Equilateral triangle</h3>The line segments making up an equilateral triangle are separated by an angle of 60°. If two vertices are on grid squares, the third must be a rotation of one of them about the other through an angle of 60°. The rotation matrix is irrational, so the rotated point must have irrational coordinates.
The math of it is this. For rotation of (x, y) counterclockwise 60° about the origin, the transformation matrix is ...
Cos(60°) is rational, but sin(60°) is not. For any non-zero rational values of x and y, the sum ...
cos(60°)·x + sin(60°)·y
will be irrational.
As in the attached diagram, if one of the coordinates of the rotated point (B) is zero, then one of the coordinates of its image (C) will be rational. The other image point coordinate cannot be rational.
