use mathematical induction to show that the statement is tru

Solution

4+6+8+...+2(n+1)=n(n+3)

Using mathematical induction

Case-1;

prove that it is true for n=1

2(1+1)=1(1+3)

2*2=1*4

4=4

so, this is true for n=1

Case-2:Assume that it is true for n=k

4+6+8+...+2(k+1)=k(k+3)

Case-3: prove that it is true for k=n+1

replace k as n+1 on left side

we get

Left side:

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

we know that

4+6+8+..+2(n+1) = n(n+3)

so, we can replace and we get

4+6+8+..+2(n+1) +2(n+1+1) = n(n+3) +2(n+1+1)

4+6+8+..+2(n+1) +2(n+1+1) = n(n+3) +2(n+2)

4+6+8+..+2(n+1) +2(n+1+1) = n^2+3n+2n+4

4+6+8+..+2(n+1) +2(n+1+1) = n^2+5n+4

4+6+8+..+2(n+1) +2(n+1+1) = (n+1)(n+4)

Right side = (n+1)(n+1+3)

right side = (n+1)(n+4)

we can see that

left side = right side = (n+1)(n+4)......Answer