17,19,33,__,129,227

• Ans 67
  17+2=19
  19+(2+12)=33
  33+(14+20)=67
  67+(34+28)=129
  129+(62+36)=227
• 13 years agoHelpfull: Yes(10) No(20)
• 17-19 = 2 => 2^2-2
  33-19 = 14 => 4^2-2
  67-33 = 34 => 6^2-2
  129-67 = 62 =>8^2-2
  227-129 = 98 => 10^2-2
  Ans: 67
• 9 years agoHelpfull: Yes(4) No(1)
• Let's analyze the sequence:
  
  17, 19, 33, __, 129, 227
  
  We need to find the missing number.
  
  ---
  
  ### Step 1: Check differences between consecutive terms:
  
  * 19 - 17 = 2
  * 33 - 19 = 14
  * __ - 33 = ?
  * 129 - __ = ?
  * 227 - 129 = 98
  
  ---
  
  ### Step 2: Look at the pattern in differences:
  
  Known differences so far: 2, 14, ?, ?, 98
  
  Look for a pattern or relationship among these differences.
  
  ---
  
  ### Step 3: Try to see if differences follow a pattern:
  
  Try to find if the differences themselves increase by a certain rule.
  
  Check the differences between the differences:
  
  14 - 2 = 12
  ? - 14 = ?
  ? - ? = ?
  98 - ? = ?
  
  Not enough info yet.
  
  ---
  
  ### Step 4: Try another approach — ratio or multiplicative pattern
  
  * 19/17 ≈ 1.12
  * 33/19 ≈ 1.74
  * 129/__ = ?
  * 227/129 ≈ 1.76
  
  Close ratios between 19 to 33 and 129 to 227.
  
  ---
  
  ### Step 5: Test if sequence can be generated by adding consecutive primes or special numbers:
  
  Look at the sequence:
  
  17 (prime)
  19 (prime)
  33 (not prime)
  ?
  129
  227
  
  ---
  
  ### Step 6: Check difference pattern in more detail:
  
  Try to find the differences more carefully:
  
  * 19 - 17 = 2 (prime)
  * 33 - 19 = 14 (not prime)
  * ? - 33 = ?
  * 129 - ? = ?
  * 227 - 129 = 98
  
  Try to see if differences are doubling:
  
  2 → 14 (7 times)
  14 → ?
  ? → ?
  ? → 98
  
  98 is 7 times 14.
  
  So maybe differences are multiplying by 7?
  
  Then the difference between ? and 33 would be 14 * 7 = 98
  But 98 is already between 129 and 227, so that may not fit.
  
  ---
  
  ### Step 7: Try to break down 129 and 227:
  
  129 - 33 = 96
  227 - 129 = 98
  
  96 and 98 are close.
  
  ---
  
  ### Step 8: Consider difference sequence as: 2, 14, 48, 96, 98
  
  Try to test if 48 fits:
  
  33 + 48 = 81
  129 - 81 = 48 (not 96)
  
  No, 48 is inconsistent.
  
  ---
  
  ### Step 9: Another way: Guess missing number is 65
  
  Test if difference fits:
  
  33 to 65 = 32
  65 to 129 = 64
  129 to 227 = 98
  
  Differences: 2, 14, 32, 64, 98
  
  No clear pattern.
  
  ---
  
  ### Step 10: Test differences in powers of 2 or multiples:
  
  Try powers of 2 multiplied by something:
  
  2 = 2 * 1
  14 = 2 * 7
  32 = 2 * 16
  64 = 2 * 32
  98 ≈ 2 * 49
  
  Try to find a pattern in multiplier: 1,7,16,32,49 (no obvious pattern)
  
  ---
  
  ### Step 11: Look again if numbers fit a quadratic or cubic formula.
  
  Check if sequence fits polynomial formula:
  
  Try general quadratic: an^2 + bn + c
  
  Positions (n): 1, 2, 3, 4, 5, 6
  Values: 17, 19, 33, x, 129, 227
  
  Set up equations for 3 points to solve for a, b, c:
  
  At n=1: a + b + c = 17
  At n=2: 4a + 2b + c = 19
  At n=3: 9a + 3b + c = 33
  
  Subtract first from second: 3a + b = 2
  Subtract first from third: 8a + 2b = 16 → 4a + b = 8
  
  From above:
  3a + b = 2
  4a + b = 8
  
  Subtracting: (4a + b) - (3a + b) = 8 - 2 → a = 6
  
  Put a=6 into 3a + b = 2 → 18 + b = 2 → b = -16
  
  Put a, b into first equation: 6 - 16 + c = 17 → c = 27
  
  ---
  
  Test n=4:
  
  Value = 6*(4^2) -16*4 + 27 = 6*16 - 64 + 27 = 96 - 64 + 27 = 59
  
  Check n=5:
  
  6*25 - 80 + 27 = 150 - 80 + 27 = 97
  
  But expected is 129, so doesn't fit.
  
  ---
  
  ### Step 12: Try cubic polynomial (an^3 + bn^2 + cn + d)
  
  Four unknowns, use four points: n=1,2,3,5
  
  At n=1: a + b + c + d = 17
  At n=2: 8a + 4b + 2c + d = 19
  At n=3: 27a + 9b + 3c + d = 33
  At n=5: 125a + 25b + 5c + d = 129
  
  Solve system to find a,b,c,d, then calculate n=4.
  
  ---
  
  Let's solve:
  
  Equation 1: a + b + c + d = 17
  Equation 2: 8a + 4b + 2c + d = 19
  Equation 3: 27a + 9b + 3c + d = 33
  Equation 4: 125a + 25b + 5c + d = 129
  
  Subtract Eq1 from Eq2:
  (8a - a) + (4b - b) + (2c - c) + (d - d) = 19 - 17
  7a + 3b + c = 2 → Eq5
  
  Subtract Eq1 from Eq3:
  (27a - a) + (9b - b) + (3c - c) + (d - d) = 33 - 17
  26a + 8b + 2c = 16 → Eq6
  
  Subtract Eq1 from Eq4:
  124a + 24b + 4c = 112 → Eq7
  
  ---
  
  From Eq5: c = 2 - 7a - 3b
  
  Put into Eq6:
  
  26a + 8b + 2(2 - 7a - 3b) = 16
  26a + 8b + 4 - 14a - 6b = 16
  (26a - 14a) + (8b - 6b) + 4 = 16
  12a + 2b + 4 = 16
  12a + 2b = 12
  6a + b = 6 → Eq8
  
  Put c into Eq7:
  
  124a + 24b + 4(2 - 7a - 3b) = 112
  124a + 24b + 8 - 28a - 12b = 112
  (124a - 28a) + (24b - 12b) + 8 = 112
  96a + 12b + 8 = 112
  96a + 12b = 104
  Divide by 4: 24a + 3b = 26 → Eq9
  
  ---
  
  Multiply Eq8 by 3:
  18a + 3b = 18 → Eq10
  
  Subtract Eq9 from Eq10:
  (18a + 3b) - (24a + 3b) = 18 - 26
  -6a = -8
  a = 8/6 = 4/3
  
  Put a=4/3 into Eq8:
  6*(4/3) + b = 6
  8 + b = 6
  b = -2
  
  Put a,b into c:
  c = 2 - 7*(4/3) - 3*(-2)
  = 2 - 28/3 + 6
  = (2 + 6) - 28/3
  = 8 - 9.33
  = -1.33 = -4/3
  
  Put a,b,c into Eq1:
  a + b + c + d = 17
  (4/3) + (-2) + (-4/3) + d = 17
  (4/3 - 4/3) + (-2) + d = 17
  0 - 2 + d = 17
  d = 19
  
  ---
  
  ### So, formula is:
  
  $T(n) = frac{4}{3}n^3 - 2n^2 - frac{4}{3}n + 19$
  
  Check n=4:
  
  $T(4) = frac{4}{3} times 64 - 2 times 16 - frac{4}{3} times 4 + 19$
  = $frac{256}{3} - 32 - frac{16}{3} + 19$
  = $frac{256 - 16}{3} - 32 + 19$
  = $frac{240}{3} - 13$
  = 80 - 13 = 67
  
  ---
  
  ### The missing number is **67**.
  
  ---
  
  ### Final sequence:
  
  17, 19, 33, **67**, 129, 227
  
  ---
  
  Want me to verify or explain more?
• 23 Days agoHelpfull: Yes(0) No(0)