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प्रश्न
Prove that the sum of the first n non-zero even numbers is n2 + n
उत्तर
Let P(n): 2 + 4 + 6 + … + 2n = n2 + n, ∀ n ∈ N
Step 1:
P(1): 2 = 12 + 1 = 2
Which is true for P(1)
Step 2:
P(k): 2 + 4 + 6 + … + 2k
= k2 + k.
Let it be true.
Step 3:
P(k + 1): 2 + 4 + 6 + … + 2k + (2k + 2)
= k2+ k + (2k + 2)
= k2 + 3k + 2
= k2 + 2k + k + 1 + 1
= (k+ 1)2 + (k + 1)
Which is true for P(k + 1)
So, P(k + 1) is true whenever P(k) is true.
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