56. The 30th term of the series 30, 25	1	, 21 , 16	1	,........ is
    	2		2

    
    
    

1. 1. 0

   
   
   

   2. - 100	1
      	2

   
   
   

   3. –183

   
   
   

   4. - 133	1
      	2

   
   
   

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1. View Hint View Answer Discuss in Forum

   Here , First term = a = 30
   

   Common difference (d) = 25	1	- 30 = - 4	1	=	- 9
   	2		2		2

   
   Number of terms = n = 30
   ∴ T_n = a + (n – 1)d
   

   ⇒ T30 = 30 + (30 – 1) ×	-9
   	2

   
   

   T30 = 30 -	29 × 9
   	2

   
   

   T30 = 30 -	261
   	2

   
   

   T30 =	60 - 261
   	2

   
   

   Correct Option: B

   Here , First term = a = 30
   

   Common difference (d) = 25	1	- 30 = - 4	1	=	- 9
   	2		2		2

   
   Number of terms = n = 30
   ∴ T_n = a + (n – 1)d
   

   ⇒ T30 = 30 + (30 – 1) ×	-9
   	2

   
   

   T30 = 30 -	29 × 9
   	2

   
   

   T30 = 30 -	261
   	2

   
   

   T30 =	60 - 261
   	2

   
   

   T30 =	- 201	= -100	1
   	2		2

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57. Find the nth term of the following sequence :
    5 + 55 + 555 + ....... T_n

1. 1. 5(10ⁿ – 1)

   
   
   

   2. 5n(10ⁿ – 1)

   
   
   

   3. 5	(10n - 1)
      9

   
   
   

   4. 	5		n	(10n - 1)
      	9

   
   
   

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1. View Hint View Answer Discuss in Forum

   Given , Series = 5 + 55 + 555 +....+ T_n
   Series = 5(1 + 11 + 111 + ..... to n terms)
   Multiplying and divided by 9 in series, we get
   

   Series =	5	(9 + 99 + 999 + ..... to n terms)
   	9

   
   

   =	5	{(10 – 1) + (10² –1) +.....+ (10n – 1)}
   	9

   
   

   Correct Option: C

   Given , Series = 5 + 55 + 555 +....+ T_n
   Series = 5(1 + 11 + 111 + ..... to n terms)
   Multiplying and divided by 9 in series, we get
   

   Series =	5	(9 + 99 + 999 + ..... to n terms)
   	9

   
   

   =	5	{(10 – 1) + (10² –1) +.....+ (10n – 1)}
   	9

   
   

   ∴ nth term =	5	(10n – 1)
   	9

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58. Find the sum of first five terms of the following series :
    

    	1	+	1	+	1	+ .......... + .........
    	1 × 4		4 × 7		7 × 10

1. 1. 9

      32

   
   
   

   2. 7

      16

   
   
   

   3. 5

      16

   
   
   

   4. 1

      210

   
   
   

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1. View Hint View Answer Discuss in Forum

   From the given question ,
   

   Expression =	1	+	1	+	1	+	1	+	1
   	1 × 4		4 × 7		7 × 10		10 × 13		13 × 16

   
   

   Expression =	1		1 -	1		+	1		1	-	1		......... +	1		1	-	1
   	3			4			3		4		7			3		13		16

   
   

   Expression =	1		1 -	1	+	1	-	1	+	1	-	1	+	1	-	1	+	1	-	1
   	3			4		4		7		7		10		10		13		13		16

   
   

   Correct Option: C

   From the given question ,
   

   Expression =	1	+	1	+	1	+	1	+	1
   	1 × 4		4 × 7		7 × 10		10 × 13		13 × 16

   
   

   Expression =	1		1 -	1		+	1		1	-	1		......... +	1		1	-	1
   	3			4			3		4		7			3		13		16

   
   

   Expression =	1		1 -	1	+	1	-	1	+	1	-	1	+	1	-	1	+	1	-	1
   	3			4		4		7		7		10		10		13		13		16

   
   

   Expression =	1		1 -	1		Expression =	1	×	15	=	5
   	3			16			3		16		16

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59. The least value of n, such that (1 + 3 + 3² + ..... + 3ⁿ) exceeds 2000, is
    

1. 1. 5
      

   
   
   

   2. 6
      

   
   
   

   3. 7
      

   
   
   

   4. 8

   
   
   

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1. View Hint View Answer Discuss in Forum

   Series ⇒ 1 + 3 + 3² +...+ 3ⁿ
   It is a geometric series whose common ratio is 3.
   As we know that ,
   

   a + ar + ar² + ...... + arn–1 =	a(rn - 1)	where , r > 1
   	r - 1

   
   Here , a = 1 , r = 3 / 1 = 3² / 3 = 3
   

   ∴ 1 + 3 + 3² +...... + 3n = Sn =	1(3n + 1 - 1)
   	3 - 1

   
   

   Sn =	3n + 1 - 1
   	2

   
   

   Correct Option: C

   Series ⇒ 1 + 3 + 3² +...+ 3ⁿ
   It is a geometric series whose common ratio is 3.
   As we know that ,
   

   a + ar + ar² + ...... + arn–1 =	a(rn - 1)	where , r > 1
   	r - 1

   
   Here , a = 1 , r = 3 / 1 = 3² / 3 = 3
   

   ∴ 1 + 3 + 3² +...... + 3n = Sn =	1(3n + 1 - 1)
   	3 - 1

   
   

   Sn =	3n + 1 - 1
   	2

   
   According to question,
   

   ∴	3n + 1 - 1	> 2000
   	2

   
   ⇒ 3ⁿ⁺¹ – 1 > 4000
   ⇒ 3ⁿ⁺¹ > 4000 + 1 = 4001
   For n = 7 , 3⁸ = 6561 > 4001

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60. The next term of the sequence,		1 +	1		;		1 +	1		;		1 +	1		;		1 +	1		;		1 +	1		;		1 +	1		_______ is.
    			2					2					3					2					3					4

1. 1. 3
      

   
   
   

   2. 	1 +	1
      		5

   
   
   

   3. 5

   
   
   

   4. 	1 +	1			1 +	1
      		2				5

   
   
   

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1. View Hint View Answer Discuss in Forum

   First term ⇒ 1 +	1	=	3
   	2		2

   
   

   Second term ⇒		1 +	1			1 +	1
   			2				3

   
   

   Second term =	3	×	4	= 2
   	2		3

   
   

   Third term ⇒		1 +	1			1 +	1			1 +	1
   			2				3				4

   
   

   Third term =	3	×	4	×	5	=	5
   	2		3		4		2

   
   

   ∴ Fourth term =		1 +	1			1 +	1			1 +	1			1 +	1
   			2				3				4				5

   
   

   Correct Option: A

   First term ⇒ 1 +	1	=	3
   	2		2

   
   

   Second term ⇒		1 +	1			1 +	1
   			2				3

   
   

   Second term =	3	×	4	= 2
   	2		3

   
   

   Third term ⇒		1 +	1			1 +	1			1 +	1
   			2				3				4

   
   

   Third term =	3	×	4	×	5	=	5
   	2		3		4		2

   
   

   ∴ Fourth term =		1 +	1			1 +	1			1 +	1			1 +	1
   			2				3				4				5

   
   

   Fourth term =	3	×	4	×	5	×	6	=	6	= 3
   	2		3		4		5		2

   
   Hence , required answer is 3.

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