Give an example of a statement P(n) which is true for all n. Justify your answer.

प्रश्न

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उत्तर

P(n) which is true for all n.

Let P(n) be,

1 + 2 + 3 + .... + n = `(n(n + 1))/2`

P(0) is 0 = `(0(0 + 1))/2` = 0; it's true.

P(1) is 1 = `(1(1 + 1))/2` = 1; it's true.

P(2) is 1 + 2 = `(2(2 + 1))/2` ; it's true.

P(k) is 1 + 2 + 3 + ... + k = `(k(k + 1))/2`.

P(k) is 1 + 2 + 3 + ... + k + 1 = `(k(k + 1))/2 + k + 1`.

= `((k + 1)(k + 2))/2`

⇒ P(k) is true for all k.

Therefore, P(n) is true for all n.

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Principle of Mathematical Induction

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अध्याय 4: Principle of Mathematical Induction - Exercise [पृष्ठ ७०]

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एनसीईआरटी एक्झांप्लर Mathematics Exemplar [English] Class 11

अध्याय 4 Principle of Mathematical Induction
Exercise | Q 2 | पृष्ठ ७०

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Principle of Mathematical Induction

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