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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

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)..

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
_________________

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

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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

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

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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)

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

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25 Apr 2017, 11:02

1

This post was 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)..

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

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

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04 Oct 2017, 23:40

1

This post was 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

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