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During the month of April, the average temperatures on day n

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During the month of April, the average temperatures on day n [#permalink]

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New post 12 Feb 2016, 23:32
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During the month of April, the average temperatures on day n in two cities A and B are denoted by TA(n) and TB(n) respectively. The average temperature in city A on any day is the average of the average temperatures of the previous two days in city A. The average temperature in city B on any day is the average of the average temperatures of the previous two days in city B. If TA(1) = 30, TA(2) = 35, TB(1) = 35 and TB(2) = 30, which of the following statements must be true?

I. TA(4) - TB(4) > TA(29) - TB(29)
II. TA(4) + TB(4) = TA(29) + TB(29)
III. |TA(n+1) - TA(n)| = |TB(n+1) - TB(n)| for n>3

A. II only
B. III only
C. I and II only
D. II and III only
E. I, II and III
[Reveal] Spoiler: OA

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Re: During the month of April, the average temperatures on day n [#permalink]

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New post 13 Feb 2016, 03:58
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TeamGMATIFY wrote:
During the month of April, the average temperatures on day n in two cities A and B are denoted by TA(n) and TB(n) respectively. The average temperature in city A on any day is the average of the average temperatures of the previous two days in city A. The average temperature in city B on any day is the average of the average temperatures of the previous two days in city B. If TA(1) = 30, TA(2) = 35, TB(1) = 35 and TB(2) = 30, which of the following statements must be true?

I. TA(4) - TB(4) > TA(29) - TB(29)
II. TA(4) + TB(4) = TA(29) + TB(29)
III. |TA(n+1) - TA(n)| = |TB(n+1) - TB(n)| for n>3

A. II only
B. III only
C. I and II only
D. II and III only
E. I, II and III


Hi,
Although its testing our visualization power to realize a certain activity will make what difference to an equation, it is a tough Q, which is not likely to be confronted, if you are not doing extremely -2 well

Now lets VISUALIZE what is happening in each case...
Important take aways, which mat be helpful in such Qs
Quote:

FIRST IMPORTANT POINT:-


The difference is 5 in first two terms and since the next terms are average of the pervious two terms, the difference in alternate terms will be based on 5..

SECOND:-


if the pervious two terms are big and small, the next will be bigger than previous..
and if the pervious two terms are small and big, the next will be smaller than previous..
so in A since first is 30 and next is 35, the next will be smaller, then bigger... etc
so even numbers will be larger than the previous odd..
in B it will be opposite...

THIRD..


in A, odd numbers will increase from 32.5 to nearly 33.25, and even numbers will decrease from 33.75 to 33.25..
in B, even numbers will increase from 31.25 to 31.75, and odd numbers will decrease from 32.5 to 31.75 ..

FOURTH


A will have no number<32.5 and B will have no number>32.5 after the two terms..

There will be many points which can clear off with the above observations too ..



but lets work out the nth term in each sequence to hit the answer straight



A.. 30, 30+5, 30+5-5/2, 30+5-5/2+5/4...
.. 30, 30+5, 30+5/2, 30+5/2+5/4,30+5/2+5/4-5/8...
.. 30, 30+5, 30+5/2, 30+15/4,30+25/8...
now lets remove 30 and work out.. 5,5/2,15/4,25/8..
5, 5*1/2, 5*3/4, 5* 5/8..
so 2nd term= 5..
3rd term= 5*1/2= 5* {2*3
-5}/2^(3-2)
4th term= 5*3/4= 5*{2*4[/b]-5}/2^(4-2)..
so nth term= 5*{2*n-5}/2^(n-2)..

so NTH term= 30 +5*{2n-5}/2^{(n-2)}

B... 35, 35-5, 35-5+5/2, 35-5+5/2-5/4..
here Nth term= 35 - 5*{2n-5}/2^{(n-2)}


let see the choices..


I. TA(4) - TB(4) > TA(29) - TB(29)
TA4= 30+5*25/4..
TB4= 35-5*25/4..
TA29= 30+5*53/2^27= nearly 30..
TB29= 35-5*53/2^27=nearly 35...

substitute values.. TRUE

II. TA(4) + TB(4) = TA(29) + TB(29)
TA4= 30+5*25/4..
TB4= 35-5*25/4..
TA29= 30+5*53/2^27= nearly 30..
TB29= 35-5*53/2^27=nearly 35...

substitute values..
30+35=30+35..
TRUE

III. |TA(n+1) - TA(n)| = |TB(n+1) - TB(n)|
it basically is asking difference in two consecutive terms
the values are equal for two sides but in opposite sign..
TRUE
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
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Re: During the month of April, the average temperatures on day n [#permalink]

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New post 18 Mar 2017, 10:05
I'd suggust approaching it differently:
1) from the given we can calculate that the 1st is true
2) second, let's look at answer choices: only C and E left, we need to test the 3rd case
3) 66,25/2-67,5/2=-0,75
63,25-62,5=0,75
hence 3rd is correct

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During the month of April, the average temperatures on day n [#permalink]

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New post 24 Mar 2017, 06:07
Hi Chetan2u,

Is the formula you mentioned is applicable for any difference in values between the terms mentioned.?
In this case the difference is 5 and hence the formula is = 5*(2n-5)/2^(n-2)
In case the difference is 3 can the formula be = 3*(2n-3)/2*(n-2)

TIA
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Re: During the month of April, the average temperatures on day n [#permalink]

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New post 25 Apr 2017, 11:02
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BOOKMARKED
chetan2u wrote:
TeamGMATIFY wrote:
During the month of April, the average temperatures on day n in two cities A and B are denoted by TA(n) and TB(n) respectively. The average temperature in city A on any day is the average of the average temperatures of the previous two days in city A. The average temperature in city B on any day is the average of the average temperatures of the previous two days in city B. If TA(1) = 30, TA(2) = 35, TB(1) = 35 and TB(2) = 30, which of the following statements must be true?

I. TA(4) - TB(4) > TA(29) - TB(29)
II. TA(4) + TB(4) = TA(29) + TB(29)
III. |TA(n+1) - TA(n)| = |TB(n+1) - TB(n)| for n>3

A. II only
B. III only
C. I and II only
D. II and III only
E. I, II and III


Hi ,
Although its testing our visualization power to realize a certain activity will make what difference to an equation, it is a tough Q, which is not likely to be confronted, if you are not doing extremely -2 well

Now lets VISUALIZE what is happening in each case...
Important take aways, which mat be helpful in such Qs
Quote:

FIRST IMPORTANT POINT:-


The difference is 5 in first two terms and since the next terms are average of the pervious two terms, the difference in alternate terms will be based on 5..

SECOND:-


if the pervious two terms are big and small, the next will be bigger than previous..
and if the pervious two terms are small and big, the next will be smaller than previous..
so in A since first is 30 and next is 35, the next will be smaller, then bigger... etc
so even numbers will be larger than the previous odd..
in B it will be opposite...

THIRD..


in A, odd numbers will increase from 32.5 to nearly 33.25, and even numbers will decrease from 33.75 to 33.25..
in B, even numbers will increase from 31.25 to 31.75, and odd numbers will decrease from 32.5 to 31.75 ..

FOURTH


A will have no number<32.5 and B will have no number>32.5 after the two terms..

There will be many points which can clear off with the above observations too ..



but lets work out the nth term in each sequence to hit the answer straight



A.. 30, 30+5, 30+5-5/2, 30+5-5/2+5/4...
.. 30, 30+5, 30+5/2, 30+5/2+5/4,30+5/2+5/4-5/8...
.. 30, 30+5, 30+5/2, 30+15/4,30+25/8...
now lets remove 30 and work out.. 5,5/2,15/4,25/8..
5, 5*1/2, 5*3/4, 5* 5/8..
so 2nd term= 5..
3rd term= 5*1/2= 5* {2*3
-5}/2^(3-2)
4th term= 5*3/4= 5*{2*4[/b]-5}/2^(4-2)..
so nth term= 5*{2*n-5}/2^(n-2)..

so NTH term= 30 +5*{2n-5}/2^{(n-2)}

B... 35, 35-5, 35-5+5/2, 35-5+5/2-5/4..
here Nth term= 35 - 5*{2n-5}/2^{(n-2)}


let see the choices..


I. TA(4) - TB(4) > TA(29) - TB(29)
TA4= 30+5*25/4..
TB4= 35-5*25/4..
TA29= 30+5*53/2^27= nearly 30..
TB29= 35-5*53/2^27=nearly 35...

substitute values.. TRUE

II. TA(4) + TB(4) = TA(29) + TB(29)
TA4= 30+5*25/4..
TB4= 35-5*25/4..
TA29= 30+5*53/2^27= nearly 30..
TB29= 35-5*53/2^27=nearly 35...

substitute values..
30+35=30+35..
TRUE

III. |TA(n+1) - TA(n)| = |TB(n+1) - TB(n)|
it basically is asking difference in two consecutive terms
the values are equal for two sides but in opposite sign..
TRUE


Hi chetan2u ,

Please explain how did you think in this way. Thanks

30, 30+5, 30+5-5/2, 30+5-5/2+5/4...
.. 30, 30+5, 30+5/2, 30+5/2+5/4,30+5/2+5/4-5/8...
.. 30, 30+5, 30+5/2, 30+15/4,30+25/8...

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During the month of April, the average temperatures on day n [#permalink]

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New post 04 Oct 2017, 23:40
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BOOKMARKED
I solved it like this, still time consuming one

A:

day 1 temp: 60/2 , diff from prev: 0
day 2 temp: 70/2 = (140/4) , diff from prev: +10/(2)
day 3 temp: 130/4 = (260/8) , diff from prev: -10/ (2 ^ 2)
day 4 temp: 270/8 = (540/16) , diff from prev: +10/(2^3)
.
. and goes on

So the point is diff between (n th) and (n-1 th) day is -------------------------(1)
+10/(2^(n-1)) => if n is even
-10/(2^(n-1)) => if n is odd


Now B:
day 1 temp: 70/2 , diff from prev: 0
day 2 temp: 60/2 = (120/4) , diff from prev: -10/(2)
day 3 temp: 130/4 = (260/8) , diff from prev: +10/ (2 ^ 2)
day 4 temp: 250/8 = (500/16) , diff from prev: -10/(2^3)
day 5 temp: 510/16 , diff from prev: +10/(2^4)
.
. and goes on

So the point is diff between (n th) and (n-1 th) day is -------------------------(1)
-10/(2^(n-1)) => if n is even
+10/(2^(n-1)) => if n is odd


TA(4) = 30 + (10/2) - (10/2^2) + (10/2^3)
TA(29) = 30 + (10/2) - (10/2^2) + (10/2^3) - ...................................... -(10/2^28)

TB(4) = 35 - (10/2) + (10/2^2) - (10/2^3)
TB(29) = 35 - (10/2) + (10/2^2) - (10/2^3) - ...................................... +(10/2^28)

Let a = (10/2) - (10/2^2) + (10/2^3),
b = (10/2) - (10/2^2) + (10/2^3) - (10/2^4)...................................... -(10/2^28)

Also let us check if a > b,
(10/2) - (10/2^2) + (10/2^3) > (10/2) - (10/2^2) + (10/2^3) - (10/2^4)...................................... -(10/2^28)
=> 0 > - (10/2^4) + (10/2^5) - ....................... -(10/2^28) => this is definitely true => so a > b ----------------------------------- eqn (2)

so TA(4) = 30 + a, TA(29) = 30 + b
TB(4) = 35 - a , TB(29) = 35 - b

Now let us see the answer choices

I. TA(4) - TB(4) > TA(29) - TB(29)

from above, (30 + a) - (35 - a) > (30 + b) - (35 - b)
=> rearranging => 2a > 2b => a > b
from eqn -----------------(2) => a > b
So this choice is true

II. TA(4) + TB(4) = TA(29) + TB(29)
from above, (30 + a) + (35 - a) = (30 + b) + (35 - b) => rearranging 65 = 65 => which is true.
so this choice also true

III. |TA(n+1) - TA(n)| = |TB(n+1) - TB(n)| for n>3


From above eqn ------------------(1),
for all n for both the cities, the absolute of diff between two days , |10/2^(n-1)|.
So this choice also true

Hence E

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During the month of April, the average temperatures on day n   [#permalink] 04 Oct 2017, 23:40
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During the month of April, the average temperatures on day n

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