11x1 t14 09 mathematical induction 2 (2011)
TRANSCRIPT
Mathematical Induction
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
which is divisible by 3
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,321 .. PPkkkei
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,321 .. PPkkkei
Step 3: Prove the result is true for n = k + 1
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,321 .. PPkkkei
Step 3: Prove the result is true for n = k + 1
integeran is where,3321 :Prove .. QQkkkei
Proof:
Proof: 321 kkk
Proof: 321 kkk
21321 kkkkk
Proof: 321 kkk
21321 kkkkk
2133 kkP
Proof: 321 kkk
21321 kkkkk
2133 kkP
213 kkP
Proof: 321 kkk
21321 kkkkk
2133 kkP
213 kkP
integeran is which 21 where,3 kkPQQ
Proof: 321 kkk
21321 kkkkk
Hence the result is true for n = k + 1 if it is also true for n = k
2133 kkP
213 kkP
integeran is which 21 where,3 kkPQQ
Proof: 321 kkk
21321 kkkkk
Hence the result is true for n = k + 1 if it is also true for n = k
2133 kkP
213 kkP
integeran is which 21 where,3 kkPQQ
Step 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction.
5by divisible is 23 Prove 23 nniv
5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
which is divisible by 5
5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
Hence the result is true for n = 1
which is divisible by 5
5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,523 .. 23 PPei kk
which is divisible by 5
5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,523 .. 23 PPei kk
which is divisible by 5
Step 3: Prove the result is true for n = k + 1
integeran is where,523 :Prove .. 333 QQei kk
Proof:
Proof: 333 23 kk
Proof: 333 23 kk
33 2327 kk
Proof: 333 23 kk
33 2327 kk
32 22527 kkP
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
integeran is which 2527 where,5 2 kPQQ
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
integeran is which 2527 where,5 2 kPQQ
Hence the result is true for n = k + 1 if it is also true for n = k
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
integeran is which 2527 where,5 2 kPQQ
Hence the result is true for n = k + 1 if it is also true for n = k
Step 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction.
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
integeran is which 2527 where,5 2 kPQQ
Hence the result is true for n = k + 1 if it is also true for n = k
Step 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction.
Exercise 6N;
3bdf, 4 ab(i), 9