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#127 Re: Help Me ! » Distance Optimization » 2015-01-21 02:06:02
#128 Re: Help Me ! » Infintite sieries » 2015-01-21 01:25:10
1) Let S be the sum of a finite geometric series with negative common ratio whose first and last terms are 1 and 4, respectively. (For example, one such series is 1-2+4, whose sum is 3.)
There is a real number L such that S must be greater than L, but we can make S as close as we wish to L by choosing the number of terms in the series appropriately. Determine L.
The series clearly has an odd number of terms. Hence
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And since n is odd,
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Thus
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Here I cheated and used WolframAlpha. 
It turns out that S_n is a decreasing sequence converging to 5/2. Hence
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#129 Re: Help Me ! » infinite arithmetic series » 2015-01-17 07:33:18
#130 Re: Help Me ! » Geo problems! » 2015-01-15 07:26:21
3) Let ABCD be a square of side length 4. Let M be on side BC such that CM = 1, and let N be on side AD such that DN = 1. We draw the quarter-circle centered at A.
Let x and y denote the areas of the shaded regions, as shown. Find x - y.
Let z be the area of the quarter-circle bounded by DN, MN and the arc. Then
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It should be straightforward to find x−y.
#131 Re: Help Me ! » What is this series and what are its next numbers? » 2015-01-15 05:21:35
If you don’t need to simplify your polynomial formula, you can write it down immediately.
15, 35, 53, 100, 243, ...
Answer:
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Can you see the trick? If you substitute n=1, all but the first term disappear; if you substitute n=2, all but the second term disappear, etc.
#132 Re: Puzzles and Games » Chess (Flash Version) » 2015-01-13 06:59:55
#133 Re: Help Me ! » Infinite Series » 2015-01-11 22:06:31
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The sum of the nth column is
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Hence
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#134 Re: Puzzles and Games » Talk in Yoda speech. » 2015-01-10 00:12:08
Object–subject–verb order Yoda uses. Called anastrophe this is. An example of hyperbaton it is.
#135 Re: Help Me ! » Transformations » 2015-01-08 07:03:25
24. Under a certain enlargement (3,4) → (0,-2) and (-5,2) → (4,-1). Find the image of (-3,-8) under this enlargement.
The quickest way to solve this is probably to plot a graph, as Bob Bundy did in post #2. However, if you need to solve it algebraically, you can do this.
Let the centre of enlargement be (a,b). We apply the translation (x,y) → (x−a,y−b) to make the centre of enlargement coincide with the origin. Under the new coordinates the enlargement becomes
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Thus if k is the enlargement factor we have the following equations:
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Solve these equations for a, b, k. The image of (−3,−8) under the enlargement is ((−3−a)k+a,(−8−b)k+b).
#136 Re: Help Me ! » Polynomial eugh » 2015-01-06 23:40:35
I found that 3i, -3i, and 2 work so theres one.
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#137 Re: Euler Avenue » What had you learned today that you found interesting? » 2014-12-12 07:25:45
I learn that it is possible to make a tetradecahedron with just regular hexagons and squares.
#138 Re: Euler Avenue » Dvorak » 2014-12-12 07:03:25
I like Dvořák’s “New World” symphony.
#139 Re: Help Me ! » finding nth row total » 2014-12-12 06:41:57
Let T(n) represent the nth triangular number; then the nth row is
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Hence the sum is
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#140 Re: Help Me ! » Can anyone prove this? » 2014-11-08 06:56:41
It is well known that
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for any a > 1.
#141 Re: Help Me ! » hard algebra » 2014-11-08 01:00:22
What’s so hard about it? 
#143 Re: Help Me ! » Algebraic problemo » 2014-11-08 00:37:02
Suppose both roots are integers. Then a and b must both be integers and the discriminant is a perfect square. We have
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[/list]Thus, if we want to be a perfect square, we must have
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#144 Re: Help Me ! » Can anyone prove this? » 2014-11-07 23:43:51
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#146 Re: Help Me ! » Hard algebra » 2014-10-31 23:51:35
[EDIT: Oops, mistake. Salvaging operation in progress.]
#147 Re: Help Me ! » Question on Matrix Terminology » 2014-10-30 12:12:15
My question: what is a matrix with variables called? That is, not a matrix of coefficients that corresponds to a linear system; rather, what is a matrix that contains variables called?
Random example: Suppose a person were to encounter a matrix that looks something like this:
where x,y are real variables and x1,y1∈R. What kind of matrix is M? Not a matrix of coefficients, nor a matrix that corresponds to a linear system (I don't think?) but a... ? matrix.
That’s just wat it is, a matrix.
While I am here, does a semicolon mean "to solve for"? I have seen it in various contexts and I was wondering if det(M)=0;y would properly mean "to solve for y".
I’ve never seen anything like this before so I don’t know what it means.
#148 Re: Help Me ! » Formal Limits » 2014-10-29 04:12:06
#149 Re: Dark Discussions at Cafe Infinity » Atheism Study » 2014-10-28 00:51:39
I mention this because a lot of people I come across seem to equate atheism with non-belief in a religion. This is not true – as in the case of Buddhism. Yet in debates between science and religion the confusion with the term is all too prevalent. 
In fact such debates tend to bracket all people into just two cetegories: “Christians” and “atheists”. If you’re religious you’re a “Christian”, if you’re not religious you’re an “atheist”. Ridiculous! 
#150 Re: Dark Discussions at Cafe Infinity » Atheism Study » 2014-10-27 07:45:53
Is a Buddhist an atheist? A Buddhist has a religion but does not believe in deities.