Home > Problem of the Week > Archive List > Detail

<< Prev 10/20/2013 Next >>

A Mathematical Pattern...No Lying!

Compared to all other subject areas, patterns are such a special part of mathematics. At times, I feel guilt when noting the wealth of patterns that one can share.


Consider this sequence...can you decipher a pattern?

  • sqrt(1 + 1*2*3*4) = 5
  • sqrt(1 + 2*3*4*5) = 11
  • sqrt(1 + 3*4*5*6) = 19
  • sqrt(1 + 4*5*6*7) = 29
Predict the value of these roots before calculating them....
  • sqrt(1 + 8*9*10*11) = ?
  • sqrt(1 + 9*10*11*12) = ?
  • .....
  • sqrt(1 + 50*51*52*53) = ?
Can you express the pattern for the general case?

 


Hint: Focus on the bolded numbers....

  • sqrt(1 + 1*2*3*4) = 5
  • sqrt(1 + 2*3*4*5) = 11
  • sqrt(1 + 3*4*5*6) = 19
  • sqrt(1 + 4*5*6*7) = 29
They are the key...

 


Solution Commentary: Consider the general case: sqrt[1 + k*(k+1)*(k+2)*(k+3)]. Simplify it to reveal the desired pattern...as this is an instance when algebra erases (or reveals) all of the magic underlying the pattern.