Amir rented a dvd and it was due to be returned on 28 march. He actually returned it to the shop on 16 april. The rental shop ap
1 answer:
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Answer:
- a) 3/5·((-2)^n + 4·3^n)
- b) 3·2^n - 5^n
- c) 3·2^n + 4^n
- d) 4 - 3 n
- e) 2 + 3·(-1)^n
- f) (-3)^n·(3 - 2n)
- g) ((-2 - √19)^n·(-6 + √19) + (-2 + √19)^n·(6 + √19))/√19
Step-by-step explanation:
These homogeneous recurrence relations of degree 2 have one of two solutions. Problems a, b, c, e, g have one solution; problems d and f have a slightly different solution. The solution method is similar, up to a point.
If there is a solution of the form , then it will satisfy ...
Rearranging and dividing by , we get the quadratic ...
The quadratic formula tells us values of r that satisfy this are ...
We can call these values of r by the names r₁ and r₂.
Then, for some coefficients p and q, the solution to the recurrence relation is ...
We can find p and q by solving the initial condition equations:
These have the solution ...
_____
Using these formulas on the first recurrence relation, we get ...
a)
__
The rest of (b), (c), (e), (g) are solved in exactly the same way. A spreadsheet or graphing calculator can ease the process of finding the roots and coefficients for the given recurrence constants. (It's a matter of plugging in the numbers and doing the arithmetic.)
_____
For problems (d) and (f), the quadratic has one root with multiplicity 2. So, the formulas for p and q don't work and we must do something different. The generic solution in this case is ...
The initial condition equations are now ...
and the solutions for p and q are ...
__
Using these formulas on problem (d), we get ...
d)
__
And for problem (f), we get ...
f)
_____
<em>Comment on problem g</em>
Yes, the bases of the exponential terms are conjugate irrational numbers. When the terms are evaluated, they do resolve to rational numbers.
- To divide the triangles into these regions, you should construct the <u>perpendicular bisector</u> of each segment.
- These perpendicular bisectors intersect and divide each triangle into three regions.
- The points in each region are those closest to the vertex in that <u>region</u>.
A triangle can be defined as a two-dimensional geometric shape that comprises three (3) sides, three (3) vertices and three (3) angles only.
<h3>What is a line segment?</h3>A line segment can be defined as the part of a line in a geometric figure such as a triangle, circle, quadrilateral, etc., that is bounded by two (2) distinct points and it typically has a fixed length.
<h3>What is a perpendicular bisector?</h3>A perpendicular bisector can be defined as a type of line that bisects (divides) a line segment exactly into two (2) halves and forms an angle of 90 degrees at the point of intersection.
In this scenario, we can reasonably infer that to divide the triangles into these regions, you should construct the <u>perpendicular bisector</u> of each segment. These perpendicular bisectors intersect and divide each triangle into three regions. The points in each region are those closest to the vertex in that <u>region</u>.
Read more on perpendicular bisectors here: brainly.com/question/27948960
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