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| 1. |
(2n+7)>(n+3)² prove by mathematical induction |
| Answer» Let P (n) =(2n + 7) < (n + 3)2For n = 1{tex}P(1) = (2 \\times 1 + 7) < {(1 + 3)^2} \\Rightarrow 9 < 16{/tex}{tex}\\therefore {/tex}\xa0P ( 1) is trueLet P(n) be true for n = k{tex}\\therefore P(k) = (2k + 7) < {(k + 3)^2}{/tex}\xa0....(1)For n = k + 1\xa0P (k + 1) = 2 (k + 1) + 7 < (k + 1 + 3)2{tex} \\Rightarrow {/tex}\xa02(k + 1) + 7 < (k + 4)2From (1)2k + 7 < (k + 3)2Adding 2 on both sides2k + 7 + 2 < (k + 3)2 + 2\xa0{tex} \\Rightarrow {/tex}\xa02(k + 1) + 7 < k2 + 9 + 6k + 2{tex} \\Rightarrow {/tex}\xa02(k + 1) + 7 < k2 +6k + 11 < k2 + 8k + 162(k + 1) + 7 <(k + 4)2{tex}\\therefore {/tex}\xa0P (k + 1) is trueThus P(k) is true {tex} \\Rightarrow {/tex}\xa0P(k + 1) is trueHence by principle of mathematical induction, P (n) is true for all {tex}n \\in N{/tex}. | |