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How they putted in underlined words H 6:05 PM page 95 v Solution: Let the given statement be P(n), i.e., p(n): (211+7) < (n + It can be observed that P(n) is true for since 2.1 + = 16, which is true. Let P(k) be true for some positive integer k, i.e., We shall now prove that P(k+ I) is true whenever p(k) is true. Consider 2(k+ l) +7 +6k +9+2 2(k+ +6k +11 ow. +8k.16 [using Thus, P(k+ I) is true whenever P(k) is true. Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n. Question

N o t e : k i s a p o s i t i v e i n t e g e r h e r e k^{2} + 6 k + 11 < k^{2} + 6 k + 11 + 2 k k^{2} + 6 k + 11 < k^{2} + 8 k + 11 A g a i n k^{2} + 8 k + 11 < k^{2} + 8 k + 11 + 5 k^{2} + 8 k + 11 < k^{2} + 8 k + 16 T h e r e f o r e, k^{2} + 6 k + 11 < k^{2} + 8 k + 16

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