NCERT Solution For Class 11 Math Chapter 9 Sequences and Series Exercise 9.1

Class 11 Maths NCERT Solutions for Chapter 9 Sequences and Series Exercise 9.1

Sequences and Series Exercise 9.1 Solutions

1. Write the first five terms of the sequences whose n^th term is a_n = n(n +2) 

Solution

a_n = n(n+2) 
Substituting n = 1, 2, 3, 4, and 5, we obtain  
a₁ = 1(1 +2) = 3 
a₂ = 2(2+2) = 8 
a₃ = 3(3+2) = 15 
a₄ = 4(4 + 2) = 24
a₅ = 5(5 + 2) = 35
Therefore, the required terms are  3, 8, 15, 24, and 35.

2. Write the first five terms of the sequences whose nth term is a_n = n/(n+1) . 

Solution

a_n = n/(n+1) 
Substituting n = 1, 2, 3, 4, 5 we obtain 
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 required terms are 1/2, 2/3, 3/4, 4/5, and  5/6.

3. Write the first five terms of the sequences whose nth term is a_n = 2ⁿ  

Solution

a_n = 2ⁿ
Substituting n = 1, 2, 3, 4, 5 we obtain
a₁ = 2¹ = 2
a₂ = 2² = 4
a₃ = 2³ = 8
a₄ = 2⁴ = 16
a₅ = 2⁵ = 32
Therefore, the required terms are 2, 4, 8, 16, and 32.

4. Write the first five terms of the sequences whose n^th term is a_n = (2n-3)/6 

Solution

Substituting n = 1, 2, 3, 4, 5, we obtain 

Therefore, the required terms are -1/6, 1/6, 1/2, 5/6 and 7/6.

5. Write the first five terms of the sequences whose nth term is  a_n = (-1)^n-1 5ⁿ⁺¹ 

Solution

Substituting n = 1, 2, 3, 4, 5, we obtain  
a₁ =  (-1)^1-1 5¹⁺¹ = 5² = 25 
a₂ = (-1)^2-1 5²⁺¹ = -5³ = -125
a₃ =  (-1)^3-1 5³⁺¹ = -5⁴ = 625
a₄ = (-1)^4-1 5⁴⁺¹ = -5⁵ = -3125
a₅ = (-1)^5-1 5⁵⁺¹ = -5⁶ = 15625
Therefore, the required terms are 25, -125, 625, -3125 and 15625.

6. Write the first terms of the sequences whose n^th term is a_n = n (n² +5)/4 

Solution

Substituting n = 1, 2, 3, 4, 5 we obtain 

Therefore, the required terms are 3/2, 9/2, 21/2, 21, and 75/2.

7. Find the 17^th term in the following sequence whose n^th term is a_n = 4n - 3: a₁₇, a₂₄

Solution

Substituting n = 17 , we obtain
a₁₇ = 4(17) - 3 = 68 - 3 = 65
Substituting n = 24, we obtain
a₂₄ = 4(24) - 3 = 96 - 3 = 93

8. Find the indicated terms in each of the sequences in Exercises 7 to 10 whose nth terms are : 
a_n = n² /2ⁿ ; a₇

Solution

Here, a_n = n² /2ⁿ 
Substituting n = 7, we obtain  
a₇ = 7² /2⁷ = 49/128

9. Find the 9^th term in the following sequence whose n^th term is a_n = (-1)^n-1 n³ ; a₉ 

Solution

Substituting n = 9, we obtain  
a₉ = (-1)^9-1 (9)³ = (9)³ = 729

10. Find the 20th term in the following sequence whose nth term is a_n = n(n-2)/(n+3); a₂₀

Solution

Substituting n = 20, we obtain
a₂₀ = 20(20-2)/(20+3) = 20(18)/23 = 360/23

11. Write the first five terms of the following sequence and obtain the corresponding series: 
a₁ = 3, a_n = 3a_n+1 + 2 for all n > 1

Solution

a₁ = 3, a_n = 3a_n+1 + 2 for all n > 1 
⇒ a₂ = 3a₁ + 2 = 3(3) + 2 = 11
a₃ = 3a₂ + 2 = 3(11) + 2 = 35 
a₄ = 3a₃ + 2 = 3(35) + 2 = 107 
a₅ = 3a₄ + 2 = 3(107) + 2 = 323 
Hence, the first five terms of the sequence are 3, 11, 35, 107, and 323. 
The corresponding series is  3+11+35+107 + 323 + ....

12. Write the first five terms of the following sequence and obtain the corresponding series:
a₁ = -1, a_n = (a_n -1)/n , n ≥ 2

Solution

Hence, the first five terms of the sequence are -1, -1/2, -1/6, -1/24 and -1/120.
The corresponding series is (-1) + (-1/2) + (-1/6) + (-1/24) + (-1/120) + ...

13. Write the first five terms of the following sequence and obtain the corresponding series: 
a₁ = a₂ = 2, a_n = a_n-1 - 1, n > 2 

Solution

a₁ = a₂ = 2, a_n = a_n-1 - 1, n > 2
⇒ a₃ = a₂ - 1 = 2 - 1 = 1
a₄ = a₃ - 1 = 1 - 1 = 0
a₅ = a₄ - 1 = 0 - 1 = -1
Hence, the first five terms of the sequence are  2, 2, 1, 0, and -1.
The corresponding series is 2 + 2 +1 + 0 + (-1) + ....

14. The Fibonacci sequence is defined by 1 = a₁ = a₂ and a_n = a_{n – 1} + a_{n – 2} , n > 2.
Find (a_n+1 )/a_n , for n = 1, 2, 3, 4, 5

Solution

1 = a₁ = a₂
a_n = a_{n – 1} + a_{n – 2} , n > 2
∴ a₃ = a₂  + a₁  = 1 + 1 = 2
a₄ = a₃  + a₂  = 2 + 1 = 3
a₅ = a₄  + a₃  = 3 + 2 = 5
a₆ = a₅  + a₄  = 5 + 3 = 8