Question
If $$n$$ is a natural number, then
A.
$${1^2} + {2^2} + ..... + {n^2} < \frac{{{n^3}}}{3}$$
B.
$${1^2} + {2^2} + ..... + {n^2} = \frac{{{n^3}}}{3}$$
C.
$${1^2} + {2^2} + ..... + {n^2} > {n^3}$$
D.
$${1^2} + {2^2} + ..... + {n^2} > \frac{{{n^3}}}{3}$$
Answer :
$${1^2} + {2^2} + ..... + {n^2} > \frac{{{n^3}}}{3}$$
Solution :
By taking option $$\left( d \right),$$
When $$n = 1,$$ then $$1 > \frac{1}{3}\,\,\,\left[ {{\text{true}}} \right]$$
When $$n = 2,$$ then $$5 > \frac{8}{3},\,\,\,\left[ {{\text{true}}} \right]$$
When $$n = 3,$$ then $$14 > 9,\,\,\,\left[ {{\text{true}}} \right]$$
When $$n = 4,$$ then $$30 > \frac{{64}}{3} = 21.33\,\,\,\left[ {{\text{true}}} \right]$$