Sequence and Series MCQ

Ans. (a) 210

Explanation: In the given series, the first term is a = 1 and the common difference is d = 2.

⇒ Using the sequences and series formulas, S_n = n/2 (2a + (n - 1) d)

⇒ For the sum of 12 terms, substitute n = 12

⇒ S₁₂ = 12/2 (2(1) + (12 - 1) 3)

∴ S₁₂ =210

Ans. (c)-1

Explanation: By the recursive formula of geometric sequence,

⇒ a₉ = r a₈ = (1/3) (-9) = -3

⇒ a₁₀ = r a₉ = (1/3) (-3) = -1.

∴ Therefore, a₁₀ = -1.

Ans. (a) 320

Explanation: We know that the 20^th term is the sum of 18^th term and 19^th term.

⇒ 20^th term = 18^th term + 19^th term

⇒ 200 + 120

∴ 20^th term = 320

Ans. (d) 8

Explanation: To find average height of the students

⇒ We have, Arithmetic mean = {Sum of Observation}/{Total numbers of Observations}

⇒ (10 + 6 + 4 + 18 + 2)/5

⇒ 40/5

∴ Average heights of students = 8ft.

Ans. (d) (720)⅙

Explanation: The GM is given as (x₁ × x₂ × x₃...× x_n)1/n

⇒ (1 × 2 × 3 × 4 × 5 x 6)1/6

∴ GM = (720)⅙

Ans. (c)99

Explanation: The given sequence is 6, 9, 12, 15.....

⇒ The first term, a = 4

⇒ The common difference, d = 9 - 4 = 5

⇒ Using the sequence and series formulas, a_n = a + (n - 1) d

⇒ For the 20^th term, substitute n = 20:

⇒ a₂₀ = a + 19d = 4 + 19×5 

⇒ a₂₀ = 4 +95 

∴ a₂₀ = 99

Ans. (b) 12

Explanation: The sequence 3, 6, 9…… is in AP, common difference d = 3 and k = 4.

⇒ Here, each term of the sequence 3, 6, 9…… is multiplied by 4.

⇒ Hence, the resultant sequence is also in AP with a common difference, k × d = 3 × 4 

∴ Thus, resultant sequence = 12

Ans. (d) 405

Explanation: Using the Fibonacci series formula, we can say that the 22^st term is the sum of the 21^th term and 20^th term.

⇒ 22^st term = 21^th term + 20^th term = 125 + 140 = 265

⇒ Now, 23^rd term = 22^nd term + 21^st term 

⇒ 23^rd term = 265 + 140

∴ 23^rd term = 405

Ans. (a) 145

Explanation: In the given series, the first term is a = 1 and the common difference is d = 3.

⇒ Using the sequences and series formulas, S_n = n/2 (2a + (n - 1) d)

⇒ For the sum of 10 terms, substitute n = 10:

⇒ S₁₀ = 10/2 (2(1) + (10 - 1) 3)

∴ S₁₀ = 145

Ans. (a) 8 (8¹⁰ - 1) / (8 - 1) 

Explanation: Here, the first term is, a = 8.

⇒ The common ratio, r = 2.

⇒ Number of terms is, n = 10.

⇒ The sum of finite geometric sequence formula is, Sn = a(rⁿ - 1) / (r - 1)

∴ S₁₀ = 8 (8¹⁰ - 1) / (8 - 1)

Ans. (a) 111

Explanation: In the given series, the first term is a = 1 and the common difference is d = 2.

⇒ Using the sequences and series formulas, S_n = n/2 (2a + (n - 1) d)

⇒ For the sum of 12 terms, substitute n = 12:

⇒ S₆ = 6/2 (2(1) + (6 - 1) 7)

∴ S₆ = 111

Ans. (d) -5

Explanation: By the recursive formula of geometric sequence,

⇒ a₉ = r a₈ = (1/2) (-20) = -10

⇒ a₁₀ = r a₉ = (1/2) (-10) = -5.

⇒ Therefore, a₁₀ = -5.

Ans. (a) 700

Explanation: We know that the 20^th term is the sum of 18^th term and 19^th term.

⇒ 20^th term = 18^th term + 19^th term

⇒ 20^th term = 600 + 100

∴ 20^th term = 700

Ans. (c) 7.5

Explanation: To find average height of the students

⇒ We have, Arithmetic mean = {Sum of Observation}/{Total numbers of Observations}

⇒ (10 + 6 + 4 + 18 + 2 + 5)/6

⇒ Height= 45/5

∴ Height = 7.5ft.

Ans. (a) (5040)^1/7

Explanation: The GM is given as (x₁ × x₂ × x₃...× x_n)1/n

⇒ GM = (1 × 2 × 3 × 4 × 5 x 6 x 7)1/7

∴ GM = (5040)^1/7

Ans. (c)50

Explanation: The given sequence is 5, 10, 15, 20.....

⇒ The first term, a = 5

⇒ The common difference, d = 10 - 5 = 5

⇒ Using the sequence and series formulas, a_n = a + (n - 1) d

⇒ For the 10^th term, substitute n = 10:

⇒ a₂₀ = a + 9d 

⇒ a₂₀ = 5 + 9×5

⇒ a₂₀ = 5 + 45 

∴ a₂₀ = 50

Ans. (c) 40

Explanation: The sequence 8, 16, 24…… is in AP, common difference d = 8 and k = 5.

⇒ Here, each term of the sequence 8, 16, 24…… is multiplied by 5.

⇒ Hence, the resultant sequence is also in AP with a common difference, k × d = 8 × 5

∴ resultant = 40