The document discusses using mathematical induction to prove the formula: 3 + 5 + 7 +...+ (2k + 1) = k(k + 2) It provides the base case of p(1) and shows that it is true. It then assumes p(k) is true, and shows that p(k+1) follows by algebraic manipulations. This completes the induction proof.

1. I’d like both proofs done on the board ...
and I’d like a couple of volunteers to not
only do them, but explain the solution as
they go. It’s your chance to be the teacher!
Once both proofs are completed, I have one
additional one I’ll do for you.
So ... volunteers for #2 first ... then #3?

2. 11.5 Mathematical Induction
Day Two
Philippians 4:6-7 "Do not be anxious about anything, but in
everything, by prayer and petition, with thanksgiving, present
your requests to God. And the peace of God, which transcends
all understanding, will guard your hearts and your minds in
Christ Jesus."

3. Use Math Induction to prove:
3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )

4. Use Math Induction to prove:
3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
p (1) : 3 = 1(1+ 2 )

5. Use Math Induction to prove:
3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
p (1) : 3 = 1(1+ 2 )
3 = 1( 3)

6. Use Math Induction to prove:
3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
p (1) : 3 = 1(1+ 2 )
3 = 1( 3)
3 = 3 ∴ p (1) is true

7. Use Math Induction to prove:
3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
p (1) : 3 = 1(1+ 2 )
3 = 1( 3)
3 = 3 ∴ p (1) is true
p ( k ) : 3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )

8. Use Math Induction to prove:
3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
p (1) : 3 = 1(1+ 2 )
3 = 1( 3)
3 = 3 ∴ p (1) is true
p ( k ) : 3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
p ( k + 1) : 3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = ( k + 1) ( k + 3)

9. Use Math Induction to prove:
3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
p (1) : 3 = 1(1+ 2 )
3 = 1( 3)
3 = 3 ∴ p (1) is true
p ( k ) : 3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
p ( k + 1) : 3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = ( k + 1) ( k + 3)

10. Use Math Induction to prove:
3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
p (1) : 3 = 1(1+ 2 )
3 = 1( 3)
3 = 3 ∴ p (1) is true
p ( k ) : 3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
p ( k + 1) : 3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = ( k + 1) ( k + 3)
3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )

11. Use Math Induction to prove:
3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
p (1) : 3 = 1(1+ 2 )
3 = 1( 3)
3 = 3 ∴ p (1) is true
p ( k ) : 3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
p ( k + 1) : 3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = ( k + 1) ( k + 3)
3 + 5 + 7 +K + ( 2k + 1) = k ( k + 2 )
3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = k ( k + 2 ) + ( 2k + 3)
continued on next slide

12. seeking:
p ( k + 1) : 3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = ( k + 1) ( k + 3)
3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = k ( k + 2 ) + ( 2k + 3)

13. seeking:
p ( k + 1) : 3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = ( k + 1) ( k + 3)
3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = k ( k + 2 ) + ( 2k + 3)
2
3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = k + 2k + 2k + 3

14. seeking:
p ( k + 1) : 3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = ( k + 1) ( k + 3)
3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = k ( k + 2 ) + ( 2k + 3)
2
3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = k + 2k + 2k + 3
2
3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = k + 4k + 3

15. seeking:
p ( k + 1) : 3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = ( k + 1) ( k + 3)
3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = k ( k + 2 ) + ( 2k + 3)
2
3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = k + 2k + 2k + 3
2
3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = k + 4k + 3
3 + 5 + 7 +K + ( 2k + 1) + ( 2k + 3) = ( k + 1) ( k + 3)
QED

16. HW #10
“I’m a great believer in luck, and I find the harder I
work the more I have of it.” Thomas Jefferson