11类NCERT解决方案-第9章序列和序列–练习9.1

^{问题1.写出第n个}项为_n = n(n + 2)的序列的前五个项。

解决方案：

Given, a_n = n(n + 2)

Putting value of n as 1, 2, 3, 4 and 5. We get,

a₁=1(1+2)=1(3)=3

a₂=2(2+2)=2(4)=8

a₃=3(3+2)=3(5)=15

a₄=4(4+2)=4(6)=24

a₅=5(5+2)=5(7)=35

Therefore, the first 5 terms of given series are 3, 8, 15, 24 and 35.

问题2.写出第n^个项为_n = n /(n +1)的级数的前5个项。

解决方案：

Given, a_n = n/(n+1)

Putting values of n as 1,2,3,4 and 5. We get,

a₁=1/(1+1)=1/2

a₂=2/(2+1)=2/3

a₃=3/(3+1)=3/4

a₄=4/(4+1)=4/5

a₅=5/(5+1)=5/6

Therefore, the first 5 terms of the given series are 1/2 , 2/3 , 3/4 , 4/5 and 5/6.

^{问题3.写出第n个}项为_n = 2 ⁿ的级数的前五个项。

解决方案：

Given, a_n=2ⁿ

Putting value of n as 1, 2, 3, 4 and 5. We get,

a₁=2¹=2

a₂=2²=4

a₃=2³=8

a₄=2⁴=16

a₅=2⁵=32

Therefore, the first 5 terms of the given series are 2, 4, 8, 16 and 32.

^{问题4.写出第n个}项为_n =(2n – 3)/ 6的系列的前五个项

解决方案：

Given, a_n=(2n -3)/6

Putting value of n as 1, 2, 3, 4 and 5. We get,

a₁=(2(1) -3)/6=-1/6

a₂=(2(2) -3)/6=1/6

a₃=(2(3) -3)/6=1/2

a₄=(2(4) -3)/6=5/6

a₅=(2(5) -3)/6=7/6

Therefore, the first 5 terms of the given series are -1/6, 1/6, 1/2, 5/6 and 7/6.

问题5.写出第n^个项为_n =(-1) ^n-1 5 ^{n + 1}的序列的前5个项。

解决方案：

Given, a_n = (-1)^n-15ⁿ⁺¹

Putting values of n as 1, 2, 3, 4 and 5. We get,
a₁ = (-1)^1-15¹⁺¹=(-1)⁰5²=25

a₂ = (-1)^2-15²⁺¹=(-1)¹5³=-125

a₃ = (-1)^3-15³⁺¹=(-1)²5⁴=625

a₄ = (-1)^4-15⁴⁺¹=(-1)³5⁵=-3125

a₅ = (-1)^5-15⁵⁺¹=(-1)⁴5⁶=15625

Therefore, the first 5 terms of the series are 25, -125, 625, -3125 and 15625.

问题6.找到系列的前五个项，其^第n个项为_n = n(n ² +5)/ 4。

解决方案：

Given, a_n = n(n²+5)/4

Putting values of n as 1, 2, 3, 4 and 5. We get,

a₁ = 1(1²+5)/4=1(1+5)/4=1(6)/4=3/2

a₂ = 2(2²+5)/4=2(4+5)/4=2(9)/4=9/2

a₃ = 3(3²+5)/4=3(9+5)/4=3(14)/4=3(7)/2=21/2

a₄ = 4(4²+5)/4=4(16+5)/4=4(21)/4=21

a₅ = 5(5²+5)/4=5(25+5)/4=5(30)/4=5(15)/2=75/2

Therefore, the first 5 terms of the series are 3/2, 9/2, 21/2, 21 and 75/2.

问题7.找到^{序列的第17个}项，其第n个项给出为_n = 4n – 3。

解决方案：

Given, a_n = 4n – 3

Putting value of n as 17. We get,

a₁₇ = 4(17) -3 = 68 – 3=65

Therefore, the 17^th term is 65.

问题8.找到^{序列的第7个}项，其第n个项给出为_n = n ² /(2n)。

解决方案：

Substituting n = 7. We get,

a₇ = 7²/(2*7)=49/14=7/2.

Therefore, the 7^th term is7/2.

问题9.找到^{序列的第9个}项，其第n个项给出为_n =(-1) ^n-1 n ³ 。

解决方案：

Substituting n = 9. We get,

a₉ = (-1)^9-19³=(-1)⁸9³=729.

Therefore, the 9^th term is 729.

问题10。找到^{序列的第20个}项，其第n个项给出为_n =(n(n-2))/(n + 3)。

解决方案：

Given, a_n = (n(n-2))/(n+3)

Putting value of n as 20. We get,

a₂₀ = (20(20 – 2))/(20+3)=(20(18))/(23)=360/23

Therefore, the 20^th term is 360/23.

问题11。找到以下序列的前5个术语。对于所有n> 1，a ₁ = 3，a _n = 3a _{n-1 + 2。}

解决方案：

a₁ = 3

a₂ = 3a_2-1+2=3a₁+2=3*3+2=9+2=11

a₃ = 3a_3-1+2=3a₂+2=3*11+2=33+2=35

a₄ = 3a_4-1+2=3a₃+2=3*35+2=105+2=107

a₅ = 3a_5-1+2=3a₄+2=3*107+2=321+2=323

The first 5 terms are 3, 11, 35, 107 and 323.

问题12.写下以下序列的前5个术语。对于所有n> 1，a ₁ = -1，a _n = a _{n-1 / n。}

解决方案：

a₁=-1

a₂=a_2-1/2=a₁/2=-1/2

a₃=a_3-1/3=a₂/3=(-1/2)/3=-1/6

a₄=a_4-1/4=a₃/4=(-1/6)/4=-1/24

a₅=a_5-1/5=a₄/5=(-1/24)/5=-1/120

Therefore, the first 5 terms of the series are -1,-1/2, -1/6, -1/24 and -1/120.

问题13.写下以下序列的前5个术语。对于所有n> 2，a ₁ = a ₂ = 2，a _n = a _{n-1 -1}

解决方案：

a₁ = 2

a₂ = 2

a₃ = a_3-1-1=a₂-1=2-1=1

a₄ = a_4-1-1=a₃-1=1-1=0

a₅ = a_5-1-1=a₄-1=0-1=-1

Therefore, the first 5 terms of the series are 2, 2, 1, 0 and -1.

问题14.斐波那契数列由下式给出：1 = a ₁ = a ₂和a _n = a _n-1 + a _n-2 ，n>2。找到前5个项_{的n + 1} / a _n。

解决方案：

a₁ = 1, a₂ = 1

a₃ = a₁+a₂=1+1=2

a₄ = a₂+a₃=1+2=3

a₅ = a₃+a₄=2+3=5

a₆ = a₄+a₅=3+5=8

Value of a_n+1/a_n for,

n = 1 is a₁₊₁/a₁=a₂/a₁=1/1=1

n = 2 is a₂₊₁/a₂=a₃/a₂=2/1=2

n = 3 is a₃₊₁/a₃=a₄/a₃=3/2

n = 4 is a₄₊₁/a₄=a₅/a₄=5/3

n = 5 is a₅₊₁/a₅=a₆/a₅=8/5

Therefore, the required answer is 1, 2, 3/2, 5/3 and 8/5