第 11 课 RD Sharma 解决方案 - 第 19 章算术级数 - 练习 19.2 |设置 1

General Formula 

Tn = a + (n – 1)*d

where,

Tn is the nth term

a is the first term

d = common difference

问题 1. 查找

ⁱ ) AP 1、4、7、10 的第 10 期......

解决方案：

So we have a₁ = a = 1

d= 4-1 = 3

n= 10

So we can calculate 10^th term of AP by using general formula-

T₁₀= a + (10-1)d

= 1 + 9*3

= 28

Hence, the 10^th term is 28.

ii) 第 18^届AP √2、3√2、5√2…..

解决方案：

So we have a₁ = a = √2

d= 3√2 – √2 = 2√2

n= 18 (given in Question )

So we can calculate 18^th term of AP by using general formula-

T₁₈= a + (18-1)d

= √2 + 17*(2√2 )

= 35√2

Hence the 18^th term is 35√2.

iii) AP 13、8、3、-2…的第 n 项。

解决方案：

So we have a1 = a = 13

d= a₂ – a₁ = 8 – 13 = -5

So n^th term is

T_n= 13 + (n-1)( – 5 )

= 13 -5n + 5

= 18 – 5n

Hence the n^th term is 18-5n.

问题 2. 在 AP 中显示 a _m+n + a _mn = 2a _m 。

解决方案：

We can prove this with the help of general formula. Let’s solve LHS.

a_m+n= a + (m+n-1)d

a_m-n = a + (m-n-1)d

and a_m = a + (m-1)d

So now,

LHS = a + (m+n-1)d + a + (m-n-1)d

= 2a + (m+n-1+m-n-1)d

= 2a+ (2m-2)d

= 2(a+(m-1)d)

= 2a_m= RHS

问题 3. i) AP 的哪一项是 3, 8, 13,…… 是 248。

解决方案：

So we are given a = 3

d= 8 – 3 = 5

T_n = 248

a + (n -1)d = 248

3 + (n-1)5 = 248

(n-1)5 = 245

n-1 = 49

n =50

Hence 50^th term of this AP is 248.

ii) AP 的哪一项是 84、80、76……是 0。

解决方案：

So we are given a = 84

d= 80 – 84 = -4

Tn = 0

a + (n -1)d = 0

84 + (n-1)(-4) =0

(n-1)(-4) = -84

n-1 = 21

n =22

Hence 22^th term of this AP is 248.

iii) AP 4、9、14….. 中的哪一项是 254？

解决方案：

So we are given a = 4

d= 9 – 4 = 5

Tn = 254

a + (n -1)d = 254

4 + (n-1)5 = 254

(n-1)5 = 250

n-1 = 50

n =51

Hence 51^th term of this AP is 248.

问题 4. i) 68 是 AP 7、10、13、……的一个术语吗？

解决方案：

We are given a = 7

d= 10 – 7 = 3

We can find whether 68 is a term of this AP by finding the valid n for 68 by using general formula. If there is no valid integer n value for 68 then it will not be a term of this AP.

T_n = 68

a+ (n-1)d = 68

7 + (n-1)3 = 68

(n-1)3= 61

n-1 = 61/3

n= 64/3

Hence we can see there is no valid integer value of n for 68 so 68 is not the term of the AP.

ii) 302 是 AP 3, 8, 13, ....的一个术语吗？

解决方案：

We are given a = 3

d= 8 – 3 = 5

We can find whether 302 is a term of this AP by finding the valid n for 302 by using general formula. If there is no valid integer n value for 302 then it will not be a term of this AP.

T_n = 302

a+ (n-1)d = 302

3 + (n-1)5 = 302

(n-1)5= 299

n-1 = 299/5

n= 304/5

Hence we can see there is no valid integer value of n for 302 so 302 is not the term of the AP.

问题 5. i) 序列 24, 23¼, 22½, 22¾, ...... 中的第一个负数是什么？

解决方案：

We are given a = 24

d= 23¼ – 24 = 93/4- 24 = -3/4

So to find first negative number we can find n value from T_n < 0

a + (n- 1)d < 0

24 + (n-1)(-3/4) < 0

24 -3n/4 +3/4 < 0

99/4 – 3n/4 <0

3n/4> 99/4

By solving this inequality we found

n>33

So we can say 34th term of AP will be first negative number.

ii) 序列 12 + 8i , 11+ 6i, 10 +4i, …… 的哪一项是 (a) 纯实数 (b) 纯虚数

解决方案：

We are given a = 12 + 8i

d= (11 + 6i) -(12 + 8i) = 11 +6i – 12 – 8i =-1 – 2i

So T_n for this sequence will be

T_n = a + (n-1)d

= 12 + 8i + (n-1)(-1-2i)

= 12 + 8i -n +1 -2in + 2i

= 13 – n + 10i -2in

= 13 – n + (10 – 2n)i

(a) For T_n to be purely real, imaginary part should be equal to 0.

So we know that 10 – 2n must be 0

10 – 2n =0

so n= 5

Hence 5^th term of A.P. is purely real.

(b) For T_n to be purely imaginary, real part should be equal to 0.

So we know that 13 – n must be 0

13 – n =0

so n= 13

Hence 13th term of A.P. is purely imaginary.

问题 6. i) AP 7, 10, 13, ....., 43 中有多少项？

解决方案：

We are given a= 7

d= 10 – 7 = 3

T_n = 43

Last term of AP is 43. So if we calculate the position of 43 then we will get the terms in this AP.

a + (n-1)d =43

7 + (n-1)3 = 43

(n-1)3 = 36

n-1 = 12

n = 13

So there are total 13 terms in this AP.

ii) AP -1、-5/6、-2/3、-1/2……、10/3 中有多少项？

解决方案：

We are given a= -1

d= -5/6 – (-1) = 1-5/6 = 1/6

Tn = 10/3

Last term of AP is 10/3. So if we calculate the position of 10/3 then we will get the terms in this AP.

a + (n-1)d =10/3

-1 + (n-1)(1/6) = 10/3

(n-1)(1/6) = 13/3

n-1 = 13*6/3

n-1= 26

n = 27

So there are total 27 terms in this AP.

问题 7. AP 的第一项是 5。共同差是 3，最后一项是 80。求词数。

解决方案：

We are given

first term a₁= a= 5

common difference d= 3

T_n = 80

Last term of AP is 80. So if we calculate the position of 80 then we will get the terms in this AP.

a + (n-1)d =80

5 + (n-1)3 = 80

(n-1)3 = 75

n-1 = 75/3

n = 25+1

n = 26

So there are total 26 terms in this AP.

问题 8. AP 的第⁶和第 17^项分别是 19 和 41。找到第 40^项。

解决方案：

We are given T₆=19 and T₁₇=41.

a + 5d = 19 —– 1

a+ 16d = 41 —– 2

On solving equation 1 and 2

a + 5d – a – 16d = 19 – 41

-11d = – 22

d = 2

and a =19 – 10

a= 9

So T₄₀ = a + 39d

= 9 + 39*2

= 9 + 78

= 87

So 40^th term is 87 in this AP.

问题 9. 如果 AP 的^第9 项为零，证明其第²⁹项是第 19^项的两倍。

解决方案：

We are given 9th term of AP is 0.

a₉=a+ 8d = 0 ——1

We have to prove, a₂₉ = 2a₁₉

We know a₂₉= a+ 28d ——-2

a₁₉= a+ 18d ——-3

From equation 1 we get

a+ 8d = 0

a= -8d

So putting a = -8d in eq 2 & 3 will give us the value of a₂₉ and a₁₉

a₂₉ = -8d +28d = 20d

a₁₉ = -8d +18d = 10d

So its proved 2a₁₉ = a₂₉.

问题 10. 如果 AP 的^第10 项的 10 倍等于第 15^项的 15 倍，证明 AP 的^第25 项为零。

解决方案：

We are given 10 times the 10^th term of an AP is equal to 15 times the 15^th term.

10a₁₀ = 15a₁₅

10(a + 9d) = 15( a +14d)

10a + 90d = 15a + 210d

5a= -120d

5a + 120d= 0

a + 24d = 0

a₂₅ = 0

So its proved a₂₅ is equal 0.

问题 11. AP 的第 10 和^第18^项分别是 41 和 73，求^第26 项。

解决方案：

We are given a₁₀ = a+ 9d = 41 ——-1

a₁₈ = a+ 17d = 73 ——-2

On solving 1 & 2

a + 9d – a -17d = 41 – 73

-8d = -32

d= 4

By substituting value of d in equation 1 we get

a= 41-9*4

a= 5

So value of 26^th term can be calculated by,

a₂₆ = a + 25d

= 5 + 25*4

= 105

So the 26th term of AP is 105.

问题 12. 在某些 AP 中，第 24^项是第 10^项的两倍。证明第⁷²项是第³⁴项的两倍。

解决方案：

We are given 24^th term of an AP is twice the 10^th term.

a₂₄ = 2a₁₀

So a + 23d = 2(a + 9d)

a + 23d = 2a + 18d

a = 5d

And we know, a₃₄= a +33d

= 5d + 33d

= 38d

Similarly, a₇₂= a +71d

= 5d + 71d

= 76d

Now we can see that a₇₂ = 2a₃₄ . Hence proved 72^nd term is twice the 34^th term.