Two consecutive numbers are removed from the series of progression 1, 2, 3 , .. n. The arithmetic mean of the of the remaining numbers is 26 1/4 . The value of N is :

A. 81
B. 60
C. 50
D. Cannot be determined.

• 1st an example (1+2+3+4+5)/2=3 this is the average now let see the result by eliminating two extreme position consecutive couple...
  1>eliminate(1,2) so (3+4+5)/3=4=3+1
  2>eliminate(4,5) so (1+2+3)/3=2=3-1
  actually (1+2+...+n)/n=n+1/2
  any two consecutive numbers you eliminate,but the average always must be in the range from (n+1/2)-1 to (n+1/2)-1
  here the average after removing consecutive nos is 26.25
  and only in case of option (C),average of (1+2+....+50)50=51/2=25.5
  it covers the range 24.5 to 26.5
  so 26.25 belongs to that range
  we can also get the two consecutive nos 26.5-26.25=.25=1/4
  the 4 in the denominator signifies that if we take consecutive nos without overlapping then the 4th couple of nos should have been removed
  that is (1,2),(3,4),(5,6),(7,8)..so 7&8 have been removed :)
  
• 9 years agoHelpfull: Yes(37) No(7)
• mean= (n*(n+1)/2)/n = (n+1)/2
  now (n+1)/2=26 1/4
  solving for n,
  n=50(approx)
• 9 years agoHelpfull: Yes(36) No(15)
• Hey guys need not worry have a look:-
  First a few important facts:- AM of a series of n natural numbers will either be a whole number or number and a half.
  For Example:-
  
  AM of 1,2,3,.....10=5.5
  1,2,3....11=6
  1,2,3.......12=6.5
  1,2,3.....13=7...Similarly....
  
  Hence with number of terms the AM is increasing by .5...............#1
  
  Second fact is the range of AM which can vary in a series of n natural number after removing 2 consecutive numbers
  is +1 and -1. i.e.
  
  1,2,3.......50 is a series with AM as 25.5
  Now if we remove the first two terms i.e. 1 and 2.We will get the AM of 3,4,5....50 as 26.5
  And if we remove 49 and 50 , we will get AM of 1,2,3....48 as 24.5
  
  This means the range of variation of AM is 24.5 to 26.5.
  This is true for all sequences of n natural numbers........................#2
  
  Now coming to the problem:-
  Given that the AM after removing the two terms is 26.25 hence we have to first check in how many ranges
  of AM it will lie and here they are:-
  
  For a series with AM as 25.5 the range is 24.5.................25.5...............26.5, hence here 26.25 will lie between 25.5 and 26.5 therefore this can be one of the possible solutions.
  Similarly the other ranges are :-
  
  AM as 26:- 25...............26.......(26.25)..27
  AM as 26.5:- 25.5........(26.25)....26.5.............27.5
  AM as 27:-26....(26.25).......27................28
  
  Hence there are only four such series where this particular mean lies in the range.
  Now we can easily figure out the terms from the mean:-
  
  If AM is 25.5 then total terms will be 50...(Formula is n=(AM*2)-1 for series of n natural numbers)
  Similarly for :-
  
  AM as 26 then n=51
  AM as 26.5 then n=52
  AM as 27 then n=53.
  
  Hence there are 4 possible series where any two consecutive numbers if removed may give the mean as 26.25.
  Here in the options since 50 matches , hence 50 is the right answer.
• 9 years agoHelpfull: Yes(34) No(9)
• Ans:50; The best method to answer the question is options method. Sum of first n natural numbers = n(n+1)/2. Sum of the numbers after removing 2 consecutive numbers = 105/4 * (n-2). Now n(n+1)/2 - 105/4 * (n-2) should be sum of two consecutive numbers. 50 satisfies this condition.
• 9 years agoHelpfull: Yes(15) No(2)
• 1,2,3......n
  sum=n(n+1)/2
  average=sum/number of terms=(n(n+1))/2(n)=(n+1)/2
  since 2 numbers are removed now, the number of terms will be (n-2)
  now the new average is (n-2)+1/2
  which is (n-1)/2
  given that this new average is equal to 26 1/4
  that is (n-1)/2 = 26 1/4
  solving this we get n=53.5 which is approximately 50
• 8 years agoHelpfull: Yes(11) No(4)
• yahoo ans.
  Let numbers removed by x and x+1
  1+2+3+...n = n(n+1)/2
  Mean = (n+1)/2
  
  1+2+3... -x -(x+1) = n(n+1)/2 - x - (x+1)
  Mean = (n(n+1)/2 - x - x - 1 )/ n-2 = 105/4
  n^2 + n - 4x - 2 = (n-2)105/2
  n^2 + n - 4x - 2 - 52.5n +105 = 0
  n^2 -51.5n +103-4x=0
  n=[ 51.5 +- root(51.5^2 +16x - 412) ]/ 2
  now this root should give a value like xyz.5 so that it can combine with 51.5 and make an integer
  here, hit and trial comes into play.
  51.5^2 - 412 = 2,240.25
  2240.25 + 16x should be a perfect square
  x=1 make this a square of 47.5
  n = 2 and n = 99/2(not integer)
  n=2 is not possible. neither is n=99/2
  so keep looking for more x
  square of 48.5 - 2240.25 gives 112 which is 7*16
  so put x=7
  n = 1.5 and n=50
  n=50 is good enough
  so 1 possible answer is
  n=50, number removed 7 and 8
  
  still.. more answers are possible
  next answer comes at x=112
  n= 115/2(not integer), and no need to check further since x is coming greater than n which is not possible, and increase in x is more than increase in n, so no answers would come.
  only answer is 50 with 7 and 8 removed
• 9 years agoHelpfull: Yes(9) No(5)
• plz solve it in a simple way..
• 9 years agoHelpfull: Yes(8) No(2)
• guys.........the mean of a series with natural numbers starting from 1 is always centred around the middle term.
  Here after removing two terms the mean still is much close to 25,indicating that it is a series having around 50 natural numbers...and looking at the options there seems only one option i.e (c)..
  
  Although the approach is hit and trail...in exams it can yield faster result...but can go wrong sometimes..
• 9 years agoHelpfull: Yes(7) No(4)
• The average of n no. is (n+1)/2
  after removing two consecutive no. the average is 105/4
  but new average depends on the values that is removed Exa-3&4 or 25&26 or any.
  so i think can not be determined
• 9 years agoHelpfull: Yes(5) No(6)
• Explanation: we solve this by checking the options.
  As the final average is 105/4, initial number of pages should be 2 more than a four multiple. So in the given options, we will check option C.
  Total = n(n+1)2=50×512=1275
  Final total = 48×1054=1260
  So sum of the pages = 15. The page numbers are 7, 8
  
  Hence answer is 50 (C)
  
• 8 years agoHelpfull: Yes(3) No(5)
• by hit & trial method we have to solve this.
  n=50, (105/4)*48=1260 (48=n-2)
  total sum =50*51/2=1275; so there is difference in 15, so two consecutive numbers are 7,8.
  ans---50(c).
  
• 8 years agoHelpfull: Yes(3) No(0)
• Ans:50; The best method to answer the question is options method. Sum of first n natural numbers = n(n+1)/2. Sum of the numbers after removing 2 consecutive numbers = 105/4 * (n-2). Now n(n+1)/2 - 105/4 * (n-2) should be sum of two consecutive numbers. 50 satisfies this condition.
• 9 years agoHelpfull: Yes(2) No(1)
• Let's us consider the numbers 1,2,3,4,5.
  Their average is 3.
  Now remove (1,2), then the average of remaining numbers is 3+1 (maximum)
  Now remove (4,5), then the average of remaining numbers is 3-1 (minimum)
  Hence for 'n' numbers the average of the remaining numbers after removing any two consecutive numbers must lie in the range (n-1)/2 to (n+3)/2.
  Now consider our question the average is 26 1/4 =105/4=26.25.
  for n=50 the range is (24.5,26.5) 26.25 lies in the given range hence n=50;
  
  Finding the remaining numbers.
  Let the numbers be x and x+1
  => (n(n+1)/2 -x-x-1 )/(n-2) =105/4
  put n=50 and solve for x
  we will get x=7
  => the numbers removed are 7,8.
• 8 years agoHelpfull: Yes(2) No(1)
• After removing numbers
  New Total / no. of term = 26 1/4
  new total / no of term = 105/4
  new total = (105/4 )(n-2)
  
  since total is not in fraction
  therefore (n-2) is divisible by 4
  
  now form option n= 50 satisfy that condition
  now we have to check is 50 is the solution or not
  new total = (105/4)(50-2) = 1260
  old total = n(n+1)/2
  = 50(51)/2 = 1275
  sum of two no. which are removed = 1275-1260 = 15
  let first no. is x
  x +( x+1) = 15
  x= 7
  
  therefore two no which are removed is 7 and 8
  therefore value of n is 50
• 8 years agoHelpfull: Yes(2) No(0)
• The answer is 50. can anyone solve it in a simple way???
• 9 years agoHelpfull: Yes(1) No(0)
• anyone solve dis pls...
• 9 years agoHelpfull: Yes(0) No(0)
• Plzz solve this ,
• 9 years agoHelpfull: Yes(0) No(0)
• Ans:50; The best method to answer the question is options method. Sum of first n natural numbers = n(n+1)/2. Sum of the numbers after removing 2 consecutive numbers = 105/4 * (n-2). Now n(n+1)/2 - 105/4 * (n-2) should be sum of two consecutive numbers. 50 satisfies this condition.
• 9 years agoHelpfull: Yes(0) No(3)
• Let the two consecutive numbers be k and k+1.
  then the AM of the remaining numbers occurs in the following manner which is 26 1/4=105/4
  => ( n(n+1)/2 - (k+(k+1)) )/n-2 = 105/4 --------->(1)
  On solving the above eq. we get the following result
  => 2n^2-103n+206=8k
  say, n=50 from the given options
  => k=7
  
  now substitute both n=50 and k=7 in eq.(1)
  => ( 25*51 - (7+8) )/48 = 105/4
  => 26.25 = 26.25
  => L.H.S = R.H.S
  Therefore the two consecutive numbers are 7 and 8
  and n=50 is the required answer.
• 8 years agoHelpfull: Yes(0) No(3)
• Use a simple logic , no need for formula !!
  
  arithmetic mean is mean of n numbers , say for
  
  (1+2+3+4+5) / 5 = 3 .
  
  In our question it is said that two terms are missing and the arithmetic mean is 26.25 or 26 1/4 !!!
  so from the options the best choice for getting the mean of 26.25 would be definitely 50 .
• 8 years agoHelpfull: Yes(0) No(3)
• n+1/2=26 1/4
  n=51.5
  so,answer is d.
• 7 years agoHelpfull: Yes(0) No(2)
• 26 1/4 is very close to the average of 50 terms. so lets assume that there were 50 terms.
  then the sum of these 50 terms will be 50*51/2=1275
  and sum of the 4 terms will be 26 1/4*48=1260.
  the difference is 15.
  hence number 7 and 8 have been removed
• 6 years agoHelpfull: Yes(0) No(1)