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

Answer 1: 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.

Question 2: Write the first five terms of the sequences whose n^th term is 

Answer 2:

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

Therefore, the required terms are .

Question 3: Write the first five terms of the sequences whose n^th term is a_n = 2ⁿ

Answer 3: 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.

Question 4: Write the first five terms of the sequences whose n^th term is

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

Therefore, the required terms are

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

Answer 5: 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.

Question 6: Write the first five terms of the sequences whose n^th term is

Answer 6: Substituting n = 1, 2, 3, 4, 5, we obtain         

Therefore, the required terms are

Question 7: Find the 17th term in the following sequence whose n^th term is

a_n = 4n-3;a₁₇, a₂₄

Answer 7: Substituting n = 17, we obtain

a₁₇ = 4(17)-3 = 68-3 = 65

Substituting n = 24, we obtain

a₂₄ = 4(24)-3 = 96-3 = 93

Question 8: Find the 7^th term in the following sequence whose n^th term is

Answer 8: Substituting n = 7, we obtain

Question 9: Find the 9^th term in the following sequence whose n^th term is

a_n = (-1)^n-1 n³;a₉

Answer 9: Substituting n = 9, we obtain

a₉ = (-1)^9-1(9)³ = (9)³ = 729

Question 10: Find the 20^th term in the following sequence whose n^th term is

Answer 10: Substituting n = 20, we obtain

Question 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

Answer 11: 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 + …

Question 12: Write the first five terms of the following sequence and obtain the

corresponding series:

Answer 12:

Hence, the first five terms of the sequence are

The corresponding series is

Question 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

Answer 13: 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) + …

Question 14: The Fibonacci sequence is defined by 1 = a₁ = a₂ and a_n = a_n-1+a_n-2, n>2

Find , for n = 1, 2, 3, 4, 5

Answer 14: 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

∴ For n = 1,

For n = 2,

For n = 3,

For n = 4,

For n = 5,