Question

2+4+6+8+...+2n=n(n+1) proved with
mathematical induction plz help me

Answer 1

For k: 2+4+6+8+.......+2k=k (k+1) ......(1)

To prove: For (k+1): 2+4+6+8+...+2(k+1)=(k+1)(k+2)......(2)

Proof:-

2+4+6+8+....+2k+2(k+1)

From (1)

K (k+1)+2(k+1)

(k+2)(k+1).....(3)

From (2)&(3)

It'seems clear that p (k+1) is suitable for the given condition.

Hence by PMI given condition is true for all n belongs to N

Answer 2

Step-by-step explanation:

Base case: n=1

2=1(1+1)=2

Assume true for n=k, that is:

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

To show true for n=k+1

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

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

The equation is correct. It is just a simple arithmetic series. If you can remember a formula for an arithmetic series given by Sn=n(a1+an)2. In this case the first term is 2 the last term is 2n and the number of terms is n so we have:

Sn=n(2+2n)2=2n(1+n)2=n(n+1)