Question

5+7+9+...+(2n+3)=n (n+4) show proof by induction

Answer 1

Answer:

Given:

5+7+9+...+(2n+3)=n (n+4)

1. Check for n=1:

5=1*5- confirmed

2. Assume it is correct for n=k

3. Verify it is correct for n=k+1:

5+7+9+...+(2k+3)+2(k+1)+3=k(k+4)+2(k+1)+3

• k(k+4)+2(k+1)+3=
• k²+4k+2k+2+3=
• k²+6k+5=k²+k+5k+5=
• k(k+1)+5(k+1)=
• (k+1)(k+5)=
• (k+1)((k+1)+4)  

5+7+9+...+(2k+3)+2(k+1)+3=(k+1)((k+1)+4) Proved