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On the number line shown, the tick marks are equally spaced and the coordinates of points F and G are 3^11 and 3^12, respectively. Which of the following points has the coordinate -(3^12) ?
A. A
B. B
C. C
D. D
E. E
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Bunuel
We see that G is three times as great as F. If F = x, then G = 3x and the distance between two consecutive tick marks is 2x.
We need to identify the point that has the coordinate –(3^12), which is the opposite of the coordinate of point G and can be denoted by –3x.
Now, we have everything in simple terms to quickly identify the point in question.
E = F – 2x = x – 2x = –x [This is not the point we are looking for.]
D = E – 2x = –x – 2x = –3x [This is the point we are looking for.]
Answer: D
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averagedude23
Here's how I did it, which I think is a much simpler way.
First of all you need to determine the units between the two datapoints 3^11 and 3^12, which gives us the units between each tick mark.
You can do this by the equation:
3^12 - 3^11
= 3x3^11 - 3^11
= 3^11(3-1)
= (3^11)(2) which is our tick mark units
Then, we need to determine how many tick marks -(3^12) is away from 3^11 (since it must be one of the 5 possible tick marks to the left of F). We can do this by determining the distance between the two points, then dividing by the tick mark units (which will tell us how many tick marks to the left of our reference point).
We can do this with the equation:
(3^11 - (-3^12)) / (3^11)(2)
= (3^11 + 3^12)) / (3^11)(2)
= (3^11 + 3x3^11)) / (3^11)(2)
= (3^11(1 + 3)) / (3^11)(2)
= (3^11)(4) / (3^11)(2) - Then we can cancel out terms in the numerator and denominator
= 4/2 = 2
Therefore, the answer must be 2 tick marks left of 3^11 = D
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Hope it helps.
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ShubhamDadhe
i didnt took it as a sequence, what i did was calculate de diference betwen the 2 given numbers then substract that from the \(3^{11}\) then substract it again (as many times till i get the value they ask for, like this:
1.- Calculate the distance between 2 points (wich we will call X):
\(X = 3^{12}-3^{11}\\
X = 3^{11}*(3-1)\\
X = 2*3^{11}\)
2.- Calculate what point E was:
\(E = F - X\\
E = 3^{11}-2*3^{11}\\
E = -3^{11}\)
3.- Calculate what point D was:
\(D = E - X\\
D = -3^{11}-2*3^{11}\\
D = -3*3^{11}\\
D = -3^{12}\)
then we know\( -3^{12}\) is 2 points away from F, so its D.
Hope it helps
General Discussion
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Bunuel
As the tick marks are equally spaced, the numbers are in Arithmetic Progression
Common Difference = \( d = 3^{12} - 3^{11} = 3^{11}(3-1) = 2*3^{11}\)
\(a_n = a + (n-1)d\)
As we are finding the lower value (descending sequence), the difference is negative
\(-(3^{12}) = 3^{11} - (n-1)2*3^{11}\)
\(-(3^{12}) = 3^{11} (1 - (n-1)2\)
\(-(3^{12}) = 3^{11}(1 + 2 - 2n)\)
\(-3 = 3 - 2n\)
\(-6 = - 2n\)
\(n = 3\)
Therefore -(\(3^{12}\)), is the third term in the sequence. Hence, -(\(3^{12}\)) is two units on the left w.r.t to \(3^{11}\) i.e. at point D.
Option D
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Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
I just wanted to ask how come 3rd from the left is D?
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Anki111
Anki111 - We have found that\( -(3^{12})\) is the third term in the sequence (as \(n = 3\)), also our calculations were based on considering \(3^{11}\) as the first term.
Hence, in the sequence -(\(3^{12}\)) is at the third position on the left. The number lies on the left side as its value, \(-(3^{12})\) is lower than \(3^{11}\).
You can also visualize this way
n = 1, represents the first term in the sequence → Value = \(3^{11}\) ; indicated by point F
n = 2, represents the second term in the sequence → Value = \(3^{11} - (2*3^{11}) = -(3^{11})\) ; indicated by point E
n = 3, represents the third term in the sequence → Value = \(-(3^{12})\) ; indicated by point D
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Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
Posts: 4
I am confused at one point here. My question is how come 3^11 is the first term in this sequence? Where is it indicated in the question? Please help me understand if I am missing any concept here.
ShubhamDadhe
It's not the first term in the sequence A through G, it's the 6th, as you can see.
Don't let references to "first" term or "third" in the prior responses confuse you, because people are defining them arbitrarily.
All that matters is finding the point that satisfies the question, and that is relative to F and G.
Posted from my mobile device
I still don't understand this problem even with the explanations above. Does anyone have a better explanation?
averagedude23
You do not have to think of 3^11 and 3^12 as 3^11*3. Think of 3^12 as 3^11+3^11+3^11 (3^11 * 2). Perhaps this helps you. It certainly did help me.
Now the problem states that in the line shown, the tick marks are equally spaced. Let´s see the space bt 3^11 and 3^12
3^11- (3^11+3^11+3^11)= 2 (3^11 + 3^11).
So, all ticks are spaced by 2 (hence, by 3^11 + 3^11).
Point E is the -(3^11) and point D is 2*-(3^11) which is the opposite of 3^12.
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Bunuel
We can PLUG IN THE ANSWERS, which represent the point equal to \(-(3^{12})\).
The answer choices are in ascending order, with A the smallest and E the greatest.
Thus, we can determine the correct answer by testing at most two answer choices:
D and B
If D is too small, the correct answer is E.
If D is too big and B too small, the correct answer is C.
If D is too big and B still too big, the correct answer is A.
The distance between one tick mark and the next = G-F = \(3^{12} - 3^{11} = 3^{11}(3-1) = 2*3^{11}\)
Answer choice D --> 2 tick marks below \(3^{11}\) (point F):
\(3^{11} - 2(2*3^{11}) = 3^{11} - 4*3^{11} = 3^{11}(1-4) = 3^{11}(-3) = -(3^{12})\)
Success!
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Find the difference between \(3^{11}\) and \(3^{12}\). You get \(2 \times 3^{11}\).
Then you need to find how many segments away \(3^{11}\) (or you can choose \(3^{12}\), it doesn't matter) is from the desired \(-3^{12}\).
Set up the equation:
\[3^{11} - x \times 2 \times 3^{11} = -3^{12},\] where \(x\) will tell us the number of spaces away \(3^{11}\) is from the desired \(-3^{12}\).
Simplify, and you get \(x = 2\). So, move 2 spaces to the left from \(3^{11}\), which you'll end up on D.
Then you need to find how many segments away \(3^{11}\) (or you can choose \(3^{12}\), it doesn't matter) is from the desired \(-3^{12}\).
Set up the equation:
\[3^{11} - x \times 2 \times 3^{11} = -3^{12},\] where \(x\) will tell us the number of spaces away \(3^{11}\) is from the desired \(-3^{12}\).
Simplify, and you get \(x = 2\). So, move 2 spaces to the left from \(3^{11}\), which you'll end up on D.
Bunuel
Solving this question is a matter of observation.
\(3^{12} = 3 * 3^{11} \)
This tells us that 3^12 is 3 times as far away from 0 as is 3^11. So we know now that 0 is right between E and F. Both positive and negative numbers are symmetric about 0. So E will be -3^11 and D will be -3^12
Attachment:
Screenshot 2024-04-08 at 7.13.08 PM.png [ 28.66 KiB | Viewed 46566 times ]
Answer (D)
I have explained exponents on the number line here:
https://youtu.be/0rpppnnJNRs
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JeffTargetTestPrep
but how do you know the distance between 2 tick marks is 2x?? is this an assumption. Would really help to explain. Not sure if I am missing something here
joebonham
If F = 3^11 = x, then G = 3^12 = 3*3^11 = 3x
Positive difference between G and F = 3^12 - 3^11 = 3x - x = 2x (or 3^12 - 3^11 = 3^11(3 - 1) = 2*3^11 = 2x)
Hope it's clear.
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Bunuel
How it made sense to me :
3^12 - 3^11
= 3^11(3-1)
= (3^11)(2) --> this is the space between each tick
if you go one space back (e) you have 3^11 - 1(3^11)(2) = -3^11
if you go TWO spaces back (d) you get 3^11 - 2 (3^11)(2) = 3^11 -4(3^11) = -3^11 (4-1) = -3^11(3) = -3^12
(minus because we are going backwards, if we had to find a higher up tick it would be +)
Godspeed
This might help others like me, who found it conflicting to use the formula for arithmetic progression. I approached this question as follows (feel free to correct me):
G-F = 2*3^11, which is one tick. The ticks are spaced equally.
Imagine a sequence such as x,y,z,1,2,3. We are asked which term is 0. Now, 3-0=3, hence the 0 is located 3 ticks backwards from 3, which is z.
In this question , we are asked to find which term is -3^12.
Hence, 3^12 - (-3^12) = 2*3^11 * Answer ( since each tick is 2*3^11)
which gives us Answer = 3.
Meaning our answer is 3 ticks down from G.
Hence D.
G-F = 2*3^11, which is one tick. The ticks are spaced equally.
Imagine a sequence such as x,y,z,1,2,3. We are asked which term is 0. Now, 3-0=3, hence the 0 is located 3 ticks backwards from 3, which is z.
In this question , we are asked to find which term is -3^12.
Hence, 3^12 - (-3^12) = 2*3^11 * Answer ( since each tick is 2*3^11)
which gives us Answer = 3.
Meaning our answer is 3 ticks down from G.
Hence D.
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