Question

mathematical induction to prove the following proposition: P(n) : 3 + 5 + 7 + ….+2n+ 1 = n(2 + n) where n = 1, 2, 3

Answer 1

mathematical induction to prove the following proposition: P(n) : 3 + 5 + 7 + ….+2n+ 1 = n(2 + n) where n = 1, 2, 3

Answer 2

show \(P(1)\) holds:$$3=1(2+1)=3$$show \(P(2)\) holds:$$3+5=2(2+2)=8$$and lastly \(P(3)\) holds:$$3+5+7=3(2+3)=15$$

Answer 3

Thanks alot

Answer 4

to prove by induction we assume \(P(k)\) holds for some positive integer \(k\) giving:$$3+5+7+\dots+2k+1=k(2+k)$$consider adding \(2(k+1)+1\) to both sides:$$3+5+7+\dots+(2k+1)+(2(k+1)+1)=k(2+k)+2(k+1)+1\\3+5+7+\dots+(2k+1)+(2(k+1)+1)=k(k+1)+2(k+1)+(k+1)\\3+5+7+\dots+(2k+1)+(2(k+1)+1)=(k+1)((k+1)+2)$$so it works for \(P(k+1)\) and therefore shows for all natural numbers