Learning Mathematics through fun -Problem solving 

(1) 1  + 3 + 5 = 9 = 3 x 3 = 3²

Identify similar set of successive odd numbers whose sum gives a square.

(2n-1) + (2n+1) + (2n+3) = 6n +3 = 3(2n+1). To satisfy the condition, (2n+1) must be odd which must be a multiple of 3 and a square number.i.e., (2n+1) = 3,12,27,48,75 …….

It gives the required solution    10 +12 +14 = 36 = 6 x 6 = 6²    

                                              25 +27 + 29 = 81 = 9 x 9 = 9²   

                                               73 + 75 + 77 = 225 = 15 x 15 =  15²  

(2)  1+3+5+7 = 16 = 4 x4 = 4²

       Find out few more set of successive odd numbers whose sum gives a square

     (2n-1) + (2n+1) + (2n+3) + (2n+5) = 8(n+1)= 4 x 2(n+1)

     2(n+1) must be a square number  It is possible when n = 1,7,17,31 …….

13 +15 +17 +19 = 64 = 8x 8 = 8²   

33 + 35 + 37 + 39 = 144 = 12 x 12 = 12² 

61 + 63 + 65 + 67 = 256 = 16 x16 = 16²

(3) 3 + 5 = 8 = 2 x 2 x 2 = 2³  

Here the sum of two successive odd numbers give a cube. Can you spot out few such pairs.?

(2n-1) + (2n+1) = 4n ,To be a cube n = 2x³  or n = 2,16,54 ………. (31,33) ,(107.109) are two such apirs whose sum give cube.

31 + 33 = 64 = 4x4x4 = 4³  

107 + 109 = 216 = 6x 6 x6 = 6³

(4) What are the three successive odd numbers whose sum gives a cube ?

 (2n-1) + (2n+1) + (2n+3) = 6n +3 = 3 (2n+1). To give odd cube (2n+1) must be odd  or (2n+1) = .  9 , ,243 ------9 (2x-1)³  . It gives

7+9+11=27 = 3x3x3 = 3³  

241 + 243 + 245 = 729 = 9x9x9 = 9³