10级RD Sharma解决方案–第9章算术级数练习9.1

问题1：编写以下第n个项的每个序列的第一个项：

(i) _n = 3n + 2

解决方案：

Given:

a_n = 3n + 2

By putting n = 1, 2, 3, 4, 5 we get first five term of the sequence

a₁ = (3 × 1) + 2 = 3 + 2 = 5

a₂ = (3 × 2) + 2 = 6 + 2 = 8

a₃ = (3 × 3) + 2 = 9 + 2 = 11

a₄ = (3 × 4) + 2 = 12 + 2 = 14

a₅ = (3 × 5) + 2 = 15 + 2 = 17

∴ The first five terms are 5, 8, 11, 14, 17

(ii)a _n =(n – 2)/ 3

解决方案：

Given:

By putting n = 1, 2, 3, 4, 5 we get first five term of the sequence

 a_n = (n – 2)/3

a₁ = (1-2)/3 = -1/3

a₂ = (2 – 2)/3 = 0

a₃ = (3 – 2)/3 = 1/3

a₄ = (4 – 2)/3 = 2/3

a₅ = (5 – 2)/3 = 3/3 =1 

∴ The first five terms are -1/3, 0, 1/3, 2/3, 1

(iii)a _n = 3 ⁿ

解决方案：

Given: 

By putting n = 1, 2, 3, 4, 5 we get first five term of the sequence

a₁ = 3¹ = 3;

a₂ = 3² = 9;

a₃ = 3³  = 27 ;

a₄ = 3⁴ = 81;

a₅ = 3⁵ = 243.

∴ The first five terms are 3, 9, 27, 81, 243.

(iv)a _n =(3n – 2)/ 5

解决方案：

Given:

By putting n = 1, 2, 3, 4, 5 we get first five term of the sequence

a₁ = (3 * 1 – 2)/5 = 1/5

a₂ = (3 * 2 – 2)/5 = 4/5

a₃ = (3 * 3 – 2)/5  = 7/5

a₄ = (3* 4 – 2)/5= 10/5 =2

a₅ = (3 * 5 – 2)/5 =13/5 

∴ The first five terms are 1/5, 4/5, 7/5, 2, 13/5

(v)a _n =(-1) ⁿ 2 ⁿ

解决方案：

Given:

By putting n = 1, 2, 3, 4, 5 we get first five term of the sequence

a₁ = (-1)¹ . 2¹  = -2

a₂ = (-1)² . 2²  = 4

a₃ = (-1)³ . 2³   = -8

a₄ = (-1)⁴ . 2⁴  = 16

a₅ = (-1)⁵ . 2⁵  = -32

∴ The first five terms are -2, 4, -8, 16, -32

(vi)a _n = n(n – 2)/ 2

解决方案：

Given:

By putting n = 1, 2, 3, 4, 5 we get first five term of the sequence

a₁ = (1.( 1 – 2))/2 = -1/2

a₂ = (2.(2 – 2))/2 = 0

a₃ = (3.(3 – 2))/2 = 3/2

a₄ = (4.(4 – 2))/2 = 4

a₅ = (5.(5 – 2))/2 =15/2

∴ The first five terms are -1/2, 0, 3/2, 4, 15/2

(vii)a _n = n ² – n + 1

解决方案：

Given:

By putting n = 1, 2, 3, 4, 5  we get first five term of the sequence

a₁ = 1² – 1 + 1 = 1

a₂ = 2² – 2 + 1 = 3

a₃ = 3² – 3 + 1  = 7

a₄ = 4² – 4 + 1 = 13

a₅ = 5² – 5 + 1 = 21

∴ The first five terms are  1, 3, 7, 13, 21

(viii)a _n = 2n ² – 3n + 1

解决方案：

Given:

By putting n = 1, 2, 3, 4, 5 we get first five term of the sequence

a₁ = 2 .1² – 3.1 + 1 = 0

a₂ = 2. 2² – 3.2 + 1 = 3

a₃ = 2.3² – 3.3 + 1 = 10

a₄ = 2.4² – 3.4 + 1 = 21

a₅ = 2.5²  – 3.5 + 1 = 36

∴ The first five terms are 0, 3, 10, 21, 36

问题2：在以下第n个项中的每个序列中找到指示的项：

(i) _n = 5n – 4； ₁₂和₁₅

解决方案 ：

Given:

a_n = 5n – 4

a₁₂ = By putting n=12

a₁₂  = 5.12 – 4 = 56

a₁₅ = By putting n=15

a₁₅  = 5.15 – 4 = 71

∴ The required terms a₁₂ = 56, a₁₅ = 71

(ii)a _n =(3n – 2)/(4n + 5)，a ₇和a ₈

解决方案：

Given:

a_n = (3n – 2)/(4n + 5)

a₇ = By putting n=7

     = (3.7 – 2)/(4.7 + 5)

      =19/33

a₈ = By putting n = 8

       = (3.8 – 2)/(4.8 + 5)

       = 2 2/37

∴ The required terms a₇ = 19/33, a₈ = 22/37

(iii)a _n = n(n – 1)(n – 2)； ₅和₈

解决方案：

Given:

a_n = n(n – 1)(n – 2)

By putting n=5

a₅ = 5(5 – 1).(5 – 2) = 5.4.3 = 60

By putting n=8

a₈ = 8.(8 – 1).(8 – 2) = (8.7.6) = 336

∴ The required terms a₅= 60, a₈ = 336

(iv)a _n =(n – 1)(2 – n)(3 + n)； a ₁ ，a ₂ ，a ₃

解决方案：

Given:

a_n = (n – 1)(2 – n)(3 + n)

By putting n = 1

a₁ = (1 – 1)(2 – 1)(3 + 1) = 0.1.4 = 0

By putting n = 2

a₂ = (2 – 1)(2 – 2)(3 + 2) = 1.0.5 = 0

By putting n = 3

a₃ = (3 – 1)(2 – 3)(3 + 3) = 2.-1.6 = -12

∴ The required terms a₁= 0, a₂ = 0 and a₃ = -12

(v)a _n =(-1) ^{n .n} ； A _{3，A 5，A 8}

解决方案：

Given:

 a_n = (-1)ⁿ.n

By putting n=3

a₃ = (-1)³.3 = -1.3 = -3

By putting n=5

a₅ = (-1)⁵.5 = -1.5 = -5

By putting n=8

a₈ = (-1)⁸.8 = 1.8 = 8

∴ The required terms a₃= -3, a₅ = -5 and a₈ = 8

问题3：找到以下给出的每个序列的下五个项：

_(ⅰ)1 = _1，N =_{正- 1} + 2，N 2≥

解决方案：

By putting n = 2, 3, 4, 5, 6 we get next five term of the sequence

a₂ = a₁ + 2 = 1 + 2 = 3

a₃ = a₂ + 2 = 3 + 2 = 5

a₄ = a₃ + 2 = 5 + 2 = 7

a₅ = a₄ + 2 = 7 + 2 = 9

a₆ = a₅ + 2 = 9 + 2 = 11

∴ The next five terms are 3, 5, 7, 9, 11

(ii)a ₁ = a ₂ = 2，a _n = a _{n – 1} – 3，n> 2

解决方案：

By putting n = 3, 4, 5, 6, 7 we get next five term of the sequence

a₃ = a₂ – 3 = 2 – 3 = -1

a₄ = a₃ – 3 = -1 – 3 = -4

a₅ = a₄ – 3 = -4 – 3 = -7

a₆ = a₅ – 3 = -7 – 3 = -10

a₇ = a₆ – 3 = -10 – 3 = -13

∴ The next five terms are -1, -4, -7, -10, -13

(iii)a ₁ = -1，a _n =(a _{n – 1} )/(n)，n> = 2

解决方案：

By putting n = 2, 3, 4, 5, 6  we get next five term of the sequence

a₂ = a₁ /2 = -1/2

a₃ = a₂ /3 = (-1/2)/3 = -1/6 

a₄ = a₃/4 = (-1/6)/4 = -1/24

a₅ = a₄/5 = (-1/24)/5 = -1/120

a₆ = a₅/6 =(-1/120)/6 = -1/720

∴ The next five terms are -1/2, -1/6, -1/24, -1/120, -1/720

(iv)a ₁ = 4，a _n = 4a _{n – 1} + 3，n> 1

解决方案：

By putting n = 2, 3, 4, 5, 6 we get next five term of the sequence

a₂ = 4.a₁  + 3 = 4.4 + 3 = 19

a₃ = 4.a₂  + 3 = 4.19 + 3 = 79

a₄ = 4.a₃  + 3 = 4.79 + 3 = 319

a₅ = 4.a₄ + 3 = 4.319 + 3 = 1279

a₆ = 4.a₅ + 3 = 5.1279 + 3 = 5119

∴ The next five terms are 19, 79, 319, 1279, 5119