The series of differences between consecutive prime numbers is represented as dp1, dp2, dp3, .... dpn .Whare dp1 is the difference between the second and the first prime number. Find the sum of series when n = 23, given that the 23rd prime number is 83 ?
A. 81
B. 82
C. 83
D. 87

• 87 :
  
  dp1 = p2-p1 , dp2 = p3-p2 ,... On adding ..
  All gets cancelled ... except dp24 (89) and dp1(2) .
• 9 years agoHelpfull: Yes(28) No(11)
• Ans: 87
  
  dp1 = p2-p1 ,
  dp2 = p3-p2
  ..
  ..
  dp23=p24-p23
  Sum of series=(p2-p1)+(p3-p2)+.....(p23-p22)+(p24-p23)
  All terms get cancelled,except p1= 2 and p24
  So Sum= -p1+p24
  Sum of series=-2+89=87
• 8 years agoHelpfull: Yes(17) No(4)
• S1 = 1 when last prime number is 3
  S2 = 3 when last prime number is 5
  S3 = 5 when last prime number is 7 and so on.
  
  So the sum of the series is 2 less than the last prime number.
  So S23 = 83 - 2 = 81
• 9 years agoHelpfull: Yes(15) No(5)
• Ans is : 81
  dp1= p2-p1, dp2= p3-p2.......
  So, p2-p1 + p3-p2 + p4-p2 +.......... + p23-p22 = p23-p1 = 83 - 2 =81 :-)
• 9 years agoHelpfull: Yes(13) No(3)
• 81 is right ans
• 9 years agoHelpfull: Yes(8) No(3)
• 2(p1), 3(p2), 5(dp3), 7(p4), 11(p5), 13(p6), 17(p7), 19(p8), 23(p9), 29(p10), 31(p11), 37(p12), 41(p13), 43(p14), 47(p15), 53(p16), 59(p17), 61(p18), 67(p19), 71(p20), 73(p21), 79(p22), 83(p23)
  dp1 = p2-p1 = 3-2 =1
  dp2 = p3-p2 = 5-3=2
  ........................
  ........................
  dp24 = p24-p23 = 89-83 =6
  Sum= 87
  
• 9 years agoHelpfull: Yes(8) No(6)
• 81 is the right ans.
• 9 years agoHelpfull: Yes(7) No(2)
• consecutive prime no's-
  2,3,5,7..........83 and their differences 1,2,2,4,2,4....... so add all differences you will get 81 as a answer
  so 81 is write answer
• 9 years agoHelpfull: Yes(7) No(2)
• 2(p1), 3(p2), 5(dp3), 7(p4), 11(p5), 13(p6), 17(p7), 19(p8), 23(p9), 29(p10), 31(p11), 37(p12), 41(p13), 43(p14), 47(p15), 53(p16), 59(p17), 61(p18), 67(p19), 71(p20), 73(p21), 79(p22), 83(p23)
  dp1 = p2-p1 = 3-2 =1
  dp2 = p3-p2 = 5-3=2
  ........................
  ........................
  dp23 = p24-p23 = 89-83 =6
  Sum= 87
• 9 years agoHelpfull: Yes(6) No(4)
• 81 is right answer.
• 9 years agoHelpfull: Yes(4) No(2)
• the answer will be B. 82
  Explanation -prime numbers are always odd numbers.
  And difference between odd numbers is always even.
  Thus the sum of the differences will also be even.
• 8 years agoHelpfull: Yes(2) No(5)
• a is the correct answer
  
• 8 years agoHelpfull: Yes(2) No(0)
• We can solve this by writing down all the prime number from 2 to 83 and then finding the differences between them and add them all. This is a tedious task.
  
  Let me introduce you with a shortcut way:
  If n was 1.
  The series would have been just dp1, which is the difference between the second and the first prime numbers, 3 and 2.
  So, dp1 = 3 - 2 = 1
  
  If n was 2.
  The series would have been just dp1 + dp2
  dp2 = difference between 3rd and 2nd odd numbers, 5 and 3 = 5 - 3 = 2
  So sum of the series when n is 2 = dp1 + dp2 = 1 + 2 = 3
  
  If n was 3.
  The The series would have been just dp1 + dp2 + dp3
  dp3 = difference between 4th and 3rd odd numbers, 7 and 5 = 7 - 5 = 2
  So sum of the series when n is 3 = dp1 + dp2 + dp3= 1 + 2 + 2= 5
  
  Did you notice something ??
  S1 = 1 when last prime number is 3
  S2 = 3 when last prime number is 5
  S3 = 5 when last prime number is 7 and so on.
  
  So the sum of the series is 2 less than the last prime number.
  So S23 = 83 - 2 = 81
• 8 years agoHelpfull: Yes(1) No(1)
• solution of sayush gupta is correct as seen also becuase dp23 means p24-p23 I dont know why everyone is showing the answer given on testPot which is wrong.
• 6 years agoHelpfull: Yes(1) No(0)
• @Abhishek... how???
• 9 years agoHelpfull: Yes(0) No(0)
• by manual calculation, calculating the sum of the differences all those prime numbers is 87.
• 9 years agoHelpfull: Yes(0) No(5)
• https://en.wikipedia.org/wiki/List_of_perfect_numbers
• 8 years agoHelpfull: Yes(0) No(0)
• Explanation :
  Let 'p' denotes prime number.
  Given, p23 = 83.
  According to the question,
  dp1= p2-p1, dp2 = p3-p2 .......
  So, p2-p1 + p3-p2 + p4-p2 +.......... + p23-p22 = p23-p1 = 83 - 2 = 81.
• 6 years agoHelpfull: Yes(0) No(0)