1.

State true or false : for any integer m, m2 < m3 . Why?

Answer»

We take m = 2, 3, 4 etc. 

We see that 

When m = 2 : 

m2 = 22 = 2 x 2 = 4 and m3 = 23 = 2 x 2 x 2 = 8

Clearly, 4 < 8, i.e. m2 < m3 

When m = 3 : 

m2 = 32 = 3 x 3 = 9 and m3 = 33 = 3 x 3 x 3 = 27

Clearly, 9 < 27, i.e. m2 < m3 

When m = 4 :

m2 = 42 = 4 x 4 = 16 and m3 = 43 = 4 x 4 x 4 = 64

Clearly, 16 < 64, i.e. m2 < m3 

But when m = 1, 

m2 = 12 = 1 x 1 = 1 and m3 = 13 = 1 x 1 x 1 = 1

Then m2 < m3 

Thus we can say that for any positive integer (natural number) m > 1, m2 < m3 is true. 

Now, consider m = – 1, – 2, – 3 etc. 

When m = – 1 :

m2 = (- 1)2 = (- 1) x (- 1) = 1 and m3 = (- 1)3 = (- 1) x (- 1) x (- 1) = -1 

Clearly, 1 > – 1, i.e. m2 > m3 

When m = -2 : 

m2 = (- 2)2 = (- 2) x (- 2) = 4 and m3 = (- 2)3 = (- 2) x (- 2) x (- 2) = – 8 

Clearly, 4 > – 8, i.e. m2 > m3 

Thus we can say that for any negative integer m, m2 < m3 is false.


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