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Question
Prove the statement by using the Principle of Mathematical Induction:
2 + 4 + 6 + ... + 2n = n2 + n for all natural numbers n.
Solution
P(n) is 2 + 4 + 6 + ..... + 2n = n2 + n.
So, substituting different values for n, we get,
P(0) = 0 = 02 + 0 Which is true.
P(1) = 2 = 12 + 1 Which is true.
P(2) = 2 + 4 = 22 + 2 Which is true.
P(3) = 2 + 4 + 6 = 32 + 2 Which is true.
Let P(k) = 2 + 4 + 6 + …+ 2k = k2 + k be true;
So, we get,
⇒ P(k + 1) is 2 + 4 + 6 + … + 2k + 2(k + 1) = k2 + k + 2k + 2
= (k2 + 2k +1) + (k + 1)
= (k + 1)2 + (k + 1)
⇒ P(k + 1) is true when P(k) is true.
Therefore, by Mathematical Induction,
2 + 4 + 6 + …+ 2n = n2 + n is true for all natural numbers n.
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