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BaLLatris [955]
2 years ago
7

Find the solution of the recurrence relation an = 3an−1 −3an−2 +an−3 if a0 = 2, a1 = 2, and a2 = 4.

Mathematics
1 answer:
Shkiper50 [21]2 years ago
5 0

Using the recurrence relation, we can find a couple more values in the sequence:

  • a3 = 3a2 -3a1 +a0 = 3(4) -3(2) +2 = 8
  • a4 = 3a3 -3a2 +a1 = 3(8) -3(4) +2 = 14

First differences are 0, 2, 4, 6, ...

Second differences are constant at 2, so the function is quadratic.

The sequence can be described by the quadratic ...

... an = n² -n +2

_____

We know the value for n=0 is 2, so we can find <em>a</em> and <em>b</em> using the given values for a1 and a2.

... an = an² +bn +2

... a1 = 2 = a·1² +b·1 +2 . . . . for n=1

... a + b = 0

... a2 = 4 = a·2² -a·2 +2 . . . . for n = 2; using b=-a from the previous equation

... 2 = 2a

... a = 1 . . . . so b = -1

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1 year ago
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3 years ago
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2 years ago
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5 0
2 years ago
The perimeter of a rectangular lot of land is 436 ft. this includes an easement of x feet of uniform width inside the lot on whi
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Given the Perimeter of the Lot is 436 ft, and an inner building whose base measures 122ft by 60ft. The key in the problem is that the easement is uniform.

Let X be the measure of easement. See attached

You will derive the equation where if you add X on the length of the inner rectangle length and width, you have below equation

P = 2L + 2W
P = 2*(122+2X) + 2*(60+2x)
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Rearranging the equation by shifting all constant on one side.
8x = 436-244-120
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Answer is 9ft easement for with respect to the base of the inner building.

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2 years ago
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