Question

Prove 2+4+6+....+2n=n(n+1) using mathematical induction

Answer 1

To prove that 2+4+6+…+2n=n(n+1) using mathematical induction, we first show that the statement is true for n=1. If n=1, then 2n=2(1)=2 and n(n+1)=1(1+1)=2. Therefore, the statement is true for n=1.

Next, we assume that the statement is true for some arbitrary value of n=k. That is, we assume that 2+4+6+…+2k=k(k+1).

Finally, we show that the statement is also true for n=k+1. That is, we show that 2+4+6+…+2(k+1)=(k+1)(k+2).

Starting with the left-hand side of the equation, we have:

2+4+6+…+2k+(k+1)

Using our assumption that 2+4+6+…+2k=k(k+1), we can substitute this expression into the left-hand side of the equation:

k(k+1)+(k+1)

Factoring out (k+1), we get:

(k+1)(k+2)

Therefore, we have shown that if the statement is true for n=k, then it is also true for n=k+1. Since we have already shown that the statement is true for n=1, it follows by mathematical induction that the statement is true for all positive integers n.

Explanation:

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Answer 2

Explanation:

We will use the method of mathematical induction to solve this question.

Step 1: Show that the statement holds for a particular value of n.

Let n = 1.

2 + 4 = 2(1 + 1) = 4

So, 2 + 4 = 1(1 + 1).

Step 2: Assume that it is true for n = k, and prove that it is also true for n = k + 1.

Let k = 1.

So, 2 + 4 + 6 = 2(1 + 1) + 6 = 4 + 6 = 10

Assume, 2 + 4 + 6 + .... + 2k = k(k + 1).

Then, 2 + 4 + 6 + .. + 2k + 2(k + 1) = k(k + 1) + 2(k + 1)

= (k + 1)(k + 2).

Step 3: The statement is true for n = 1 and also for n = k + 1, hence it is true for all positive values of n.

Therefore, 2 + 4 + 6 + .... + 2n = n(n + 1).