Answer
This question doesn't have a general solution/proof. We just have to check the the given statement by substituting the values of n.
For n=1 we have,
$n^{2}$+1=2 and $2^{n}$=2, (2=2)
therefore for n=1 the statement holds.
Similarly for n=2,
$n^{2}$+1=5 and $2^{n}$=2 (5>2 therefore statement is true for n=2)
For n=3,
$n^{2}$+1=10 and $2^{n}$=8 (10>8 therefore the statement is true for n=3)
For n=4,
$n^{2}$+1=17 and $2^{n}$=16 (17>16 therefore the statement is true for n=4)
It can also be observed that the difference between the values is decreasing on increasing the value of n. Therefore after n=4 i.e, n=5,6,7,..., the statement will become false.
Work Step by Step
This question doesn't have a general solution/proof. We just have to check the the given statement by substituting the values of n.
For n=1 we have,
$n^{2}$+1=2 and $2^{n}$=2, (2=2)
therefore for n=1 the statement holds.
Similarly for n=2,
$n^{2}$+1=5 and $2^{n}$=2 (5>2 therefore statement is true for n=2)
For n=3,
$n^{2}$+1=10 and $2^{n}$=8 (10>8 therefore the statement is true for n=3)
For n=4,
$n^{2}$+1=17 and $2^{n}$=16 (17>16 therefore the statement is true for n=4)
It can also be observed that the difference between the values is decreasing on increasing the value of n. Therefore after n=4 i.e, n=5,6,7,..., the statement will become false.
