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