Statement-1 For all natural number n, `1+2+....+nlt (2n+1)^2` Statement -2 For all natural numbers , `(2n+3)^2-7(n+1)lt (2n+3)^3` .

Options

A. Statement -1 is true , Statement -2 is true Statement -2 is correct explanation for Statement -1.
B. Statement -1 is true , Statement -2 is true , Statement -2 is not the correct explanation for Statement -1
C. Statement-1 is true , Statement-2 is false
D. Statement-1 is false , Statement -2 is true .

Answer

B

Explanation

Let `P(n):1+2+3+......+n lt (2n+1)^2`
Step I For n=1,
`P(1):1lt (2+1)^2rArr 1lt 9`
which is true .
Step II Assume P(n) is true for `n=k`, then
`P(k):1+2+.....+klt (2k+1)^2`
Step III For `n=k +1`, we shall prove that
`P(k+1):1+2+3+....+k+(k+1)lt (2k+3)^2`
From assumption step
`1+2+3+....+k+(k+1)lt (2k+1)^2+k+1`
`=4k^2+5k+2`
`=(2k+3)^2-7(k+1)lt (2k+3)^2 [ because 7(k+1)gt0]`
`therefore P(k+1)` is true .
Here , both Statements are true but Statement -2 is not correct explanation of Statement -1.