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Senior Manager  Joined: 20 Aug 2015
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GMAT 1: 760 Q50 V44 During the month of April, the average temperatures on day n  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 23% (03:21) correct 77% (03:13) wrong based on 1390 sessions

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

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

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2
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
General Discussion
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During the month of April, the average temperatures on day n  [#permalink]

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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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1
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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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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Re: During the month of April, the average temperatures on day n  [#permalink]

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VeritasPrepKarishma is there an easy and quick approach to this question?
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Re: During the month of April, the average temperatures on day n  [#permalink]

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Is there any short approach for this question
KarishmaB Bunuel or chetan2u please explain.
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Re: During the month of April, the average temperatures on day n  [#permalink]

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Let us try to visualise this problem by assuming two variables in place of real values.

Let us consider TA(1)=a, TA(2)=b and therefore TB(1)=b, TB(2)=a.

Now look at the series:
TA(3) = (a/2)+(b/2)
TB(3) = (a/2)+(b/2)
TA(4) = (a/4)+(3b/4)
TB(4) = (3a/4)+(b/4)
TA(5) = (3a/8)+(5b/8)
TB(5) = (5a/8)+(3b/8)

If you look closely, coefficients of a and b are getting interchanged in both the expressions TA and TB. Also, the interchanged coefficients in TB are (1-coefficient of TA)

Therefore, if we assume TA(n) = La+Mb, then TB(n) = (1-L)a+(1-M)b; L and M are assumed coefficients of a and b respectively in nth expression.

Now, coming to options.

Option (2) = TA(n)+TB(n) = La+Mb+ (1-L)a+(1-M)b = a+b, which is constant and independent of n
Hence option(2) is correct

Option (3): |TA(n+1) - TA(n)| = |TB(n+1) - TB(n)| for n>3

Let us consider TA(n+1) = La+Mb
TA(n) = Pa+Qb

Hence, TB(n+1) = (1-L)a+(1-M)b
TB(n) = (1-P)a+(1-Q)b

Put both the values in expression of Option 3:
L.H.S = (L-P)a+(M-Q)b
R.H.S = (P-L)a+(Q-M)b

Hence |L.H.S|= |R.H.S|

Option 1:TA(4) - TB(4) > TA(29) - TB(29)
Let us consider TA(n) = La+Mb
TB(n) = (1-L)a+(1-M)b

Therefore, TA(n) - TB(n) = 2(La+Mb)-(a+b)= 2TA(n) - (a+b)

Hence, TA(n) - TB(n) is proportional to TA(n)
Please note that TA(n) decreased as n increases, therefore TA(4) - TB(4) > TA(29) - TB(29).
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Re: During the month of April, the average temperatures on day n  [#permalink]

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I didnt get the solution to this problem in the first attempt. However, on my second attempt, the underlying concept which is tested in the question became crystal clear to me.

Evaluate TA and TB upto n=4, then draw out a number line- to visualize the concept of averages.

30---32.5---35

Remove 30

0-----2.5----5

Now the no line is divided into equal parts as we keep on taking the averages:

0---1.25---2.5---3.75--5

TA4= 3.75
TB4= 1.75

Hence TA4 will keep decreasing towards 2.5 and TB4 will keep increasing towards 2.5 . Therefore the first statement can be proved- TA29 <TA4 and TB29 >TB4.

Now lets directly take statement 3.

since both Ta and TB are moving towards 2.5 in equal amounts. On any day, the difference from the previous day for both A and B will be same ( Please refer the image) and statement 2 is a direct result of statement 3.

Kudos the post if you liked my solution.

Thanks
Attachments 20180803_215738.jpg [ 3.81 MiB | Viewed 16586 times ]

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

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TA(n) --> even > odd
so ; TA(4) > TA(29)
TB(n) --> odd > even
so ; TB(4) < TB(29)

I. TA(4) - TB(4) > TA(29) - TB(29)
--> TA(4) - TA(29) > TB(4) - TB(29)
--> (+) > (-) ; True

II. TA(4) + TB(4) = TA(29) + TB(29)
--> I try if n = 3, 4
--> TA(n) + TB(n) = 65 ; True

III. |TA(n+1) - TA(n)| = |TB(n+1) - TB(n)| for n>3
--> ΔTA(n) = ΔTB(n) ; True
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Re: During the month of April, the average temperatures on day n  [#permalink]

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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

The quickest way to approach this would be graphical.

We can see the when n>3, the temperatures TA(n) and TB(n) progress symmetrically about y=32.5 and tends to flatten out

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

As for the first statement, the gap between TA(4) and TB(4) will obviously be much larger than that between TA(29) and TB(29)

So I is true

II. TA(4) + TB(4) = TA(29) + TB(29)

Since the projections are symmetrical about y=32.5, the average temperatures of TA(n) and TB(n) for any n>3 will be the same

So [TA(4)+TB(4)]/2 = [TA(29)+TB(29)]/2

=> TA(4)+TB(4) = TA(29)+TB(29)

So II is also true

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

For any n>3, the rise of TA(n) will be equal to fall of TB(n) and vice versa

So, III is also true

Posted from my mobile device
Attachments IMAG0778.jpg [ 2.11 MiB | Viewed 7343 times ]

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

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SeregaP wrote:
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

Hi,
Could you please clarify what you did with case 2.
Your analysis of case 1 and 3 are easy to follow, thank you.
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Re: During the month of April, the average temperatures on day n  [#permalink]

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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 Bunuel, Could you please advise a better way to solve this question. This question seems to be very tough so when we face these kind of questions in the exam how to approach?
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Re: During the month of April, the average temperatures on day n  [#permalink]

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The Strategy to above question lies like this-
1.Don't solve it (Based on factors like if you are not doing very well )
2.Solve it - In that case, i think this approach is good -

Since the option includes I in C and E, try to eliminate by calculating I -
Following the explanation given by chetan2u follow till end of second point and rearrange the equation as -
TB(29) - TB(4) > TA (29) -TA (4).

So I is in the answer which eliminates A,B and D
Now just check the validity of III.

I checked till 4,5 and realized why the modulus of difference will always be the same.
So E it is.

Please press kudos, if I was able to help you. Thank you. Re: During the month of April, the average temperatures on day n   [#permalink] 04 Aug 2019, 08:15
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