Chapter 12 Sequences; Induction; the Binomial Theorem

Chapter 12: Sequences; Induction; the Binomial Theorem
1 1 1 1
11. I: n = 1: = and =
1(1+ 1) 2 1+
1 2
1 1 1 1 k
II: If + + + + = , then
12 ⋅ 23 ⋅ 34 ⋅ kk ( + 1) k+
1
1 1 1 1 1
+ + + + +
1⋅2 2 ⋅3 3⋅ 4 kk ( + 1) ( k+ 1)( k+ 1 + 1)
⎡ 1 1 1 1 ⎤ 1
= ⎢ + + + + ⎥+
⎣12 ⋅ 23 ⋅ 34 ⋅ kk ( + 1) ⎦ ( k+ 1)( k+
2)
2 2
k 1 k k + 2 1 k + 2k+ 1 ( k+ 1) k+ 1 k+
1
= + = ⋅ + = = = =
k + 1 ( k + 1)( k+ 2) k+ 1 k+ 2 ( k+ 1)( k+ 2) ( k+ 1)( k+ 2) ( k+ 1)( k+ 2) k+ 2 k+ 1 + 1
Conditions I and II are satisfied; the statement is true.
( )
12. I:
1 1 1 1
n = 1: = and =
(2⋅1−1)(2⋅ 1+ 1) 3 2⋅ 1+
1 3
13. I:
1 1 1 1 k
II: If + + + + = , then
1⋅3 3⋅5 5⋅7 (2k− 1)(2k + 1) 2k+
1
1 1 1 1 1
+ + + + +
1⋅3 3⋅5 5 ⋅7 (2k − 1)(2k + 1) (2( k+ 1) − 1)(2( k+ 1) + 1)
⎡ 1 1 1 1 ⎤ 1
= ⎢ + + + + ⎥+
⎣1⋅3 3⋅5 5⋅7 (2k− 1)(2k+ 1) ⎦ (2k+ 1)(2k+
3)
k 1 k 2k+
3 1
= + = ⋅ +
2k + 1 (2k + 1)(2k+ 3) 2k+ 1 2k+ 3 (2k+ 1)(2k+
3)
2
2k + 3k+ 1 ( k+ 1)(2k+ 1) k+
1 k + 1
= = = =
(2k + 1)(2k + 3) (2k+ 1)(2k+
3) 2k
+ 3 2 k+ 1 + 1
Conditions I and II are satisfied; the statement is true.
2 1
n = 1: 1 = 1and ⋅ 1(1 + 1)(2⋅ 1+ 1) = 1
6
2 2 2 2 1
( )
II: If 1 + 2 + 3 + + k = ⋅ k( k+ 1)(2k+
1) , then
6
2 2 2 2 2 2 2 2 2 2 1
2
1 + 2 + 3 + + k + ( k+ 1) = ( 1 + 2 + 3 + + k ) + ( k+ 1) = k( k+ 1)(2k+ 1) + ( k+
1)
6
⎡1 ⎤ ⎡1 2 1 ⎤ ⎡1 2 7 ⎤ 1
2
= ( k + 1) ⎢ k(2k+ 1) + k+ 1 ( k 1) k k k 1 ( k 1) k k 1 ( k 1) 2k 7k
6
6 ⎥ = + ⎢ + + +
3 6 ⎥ = + ⎢ + +
3 6 ⎥ = + + +
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 6
1
= ⋅ ( k + 1)( k+ 2)(2 k+
3)
6
= 1 ⋅ ( k + 1)[( k+ 1) + 1][2( k+ 1) + 1]
6
Conditions I and II are satisfied; the statement is true.
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