To prove by induction that 1 + 2 + \dots + n = n(n+1)/2 for all n \ge 1 , t...

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To prove by induction that 1 + 2 + \dots + n = n(n+1)/2 for all n \ge 1, the INDUCTIVE HYPOTHESIS is the assumption that:

1 + 2 + \dots + k = k(k+1)/2 holds for an arbitrary fixed integer k \ge 1

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k(k+1)/2 is always an integer

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1 + 2 + \dots + n = n(n+1)/2 holds for all n (which is what we want to prove)

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1 + 2 + \dots + (k+1) = (k+1)(k+2)/2 holds

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