Sequences and Series: Master the Concepts with Engaging Quizzes

Question 1

Which of the following sequences converges?

Accepted Answer

an = 1/n

Answer

an = n^2

Answer

an = sin(n)

Answer

an = e^n

Question 2

Which of the following functions is represented by the Taylor series ∑ (n=0 to infinity) (x - 1)^n / n!?

Accepted Answer

e^x

Answer

sin(x)

Answer

cos(x)

Answer

ln(x)

Question 3

What is the Maclaurin series expansion for the function f(x) = ln(1 + x)?

Accepted Answer

∑ (n=0 to infinity) (-1)^n x^n / n

Answer

∑ (n=0 to infinity) x^n / n

Answer

∑ (n=0 to infinity) x^(n+1) / (n+1)

Answer

∑ (n=0 to infinity) (-1)^n x^(n+1) / (n+1)

Question 4

What is the formula for calculating the sum of the first n terms of a geometric sequence with first term (a_1), common ratio (r), and nth term (a_n)?

Accepted Answer

S_n = a_1 * (1 - r^n) / (1 - r)

Answer

S_n = a_1 * (1 + r^n) / (1 + r)

Answer

S_n = a_1 * (1 - r^n) / (1 + r)

Answer

S_n = a_1 * (1 + r^n) / (1 - r)

Question 5

Which of the following is NOT a type of sequence?

Accepted Answer

Harmonic

Answer

Arithmetic

Answer

Geometric

Answer

Fibonacci

Question 6

In the arithmetic sequence 3, 7, 11, 15, ..., what is the 10th term?

Accepted Answer

41

Answer

37

Answer

45

Answer

49

Question 7

What is the common ratio of the geometric sequence 2, 10, 50, 250, ...?

Accepted Answer

5

Answer

2

Answer

10

Answer

50

Question 8

Which of the following is a characteristic of a geometric sequence?

Accepted Answer

The terms have a constant ratio.

Answer

The terms increase by a constant amount.

Answer

The first term is always equal to 0.

Answer

The sequence can be either increasing or decreasing.

Question 9

Which of the following sequences is divergent?

Accepted Answer

0.1, 0.01, 0.001, ...

Answer

1, 1/2, 1/4, 1/8, ...

Answer

1, 2, 4, 8, 16, ...

Answer

-1, 1, -1, 1, ...

Question 10

Which of the following is NOT an application of sequences and series in real life?

Accepted Answer

Finding the probability of winning a lottery

Answer

Predicting population growth

Answer

Calculating compound interest

Answer

Modeling radioactive decay

Question 11

Which of the following is NOT a property of a geometric sequence?

Accepted Answer

The terms are all equal.

Answer

The terms have a common ratio.

Answer

The ratio of consecutive terms is constant.

Answer

The sum of the terms is always finite.

Question 12

Find the sum of the first 20 terms of the arithmetic sequence 1, 3, 5, ...

Accepted Answer

210

Answer

220

Answer

230

Answer

240

Question 13

Find the sum of the first 10 terms of the geometric sequence 1/2, 1, 2, ...

Accepted Answer

1024

Answer

1023

Answer

1025

Answer

1026

Question 14

Which of the following is NOT a property of an arithmetic sequence?

Accepted Answer

The terms are all equal.

Answer

The terms have a common difference.

Answer

The sequence is either increasing or decreasing.

Answer

The sum of the terms is always finite.

Question 15

What is the formula for the nth term of an arithmetic sequence?

Accepted Answer

an = a1 + nd

Answer

an = a1 + r^(n-1)

Answer

an = a1 * nd

Answer

an = a1 * r^(n-1)

Question 16

What is the formula for the nth term of a geometric sequence?

Accepted Answer

an = a1 * r^(n-1)

Answer

an = a1 + nd

Answer

an = a1 + r^(n-1)

Answer

an = a1 * nd

Question 17

Find the limit of the sequence an = (n+1)/(n-1).

Accepted Answer

1

Answer

0

Answer

-1

Answer

Does not exist

Question 18

Which of the following is a necessary condition for a sequence to be convergent?

Accepted Answer

The sequence must be bounded.

Answer

The sequence must be increasing.

Answer

The sequence must be decreasing.

Answer

The sequence must have a finite number of terms.

Question 19

Find the nth term (nth element) of an arithmetic sequence with the first term equal to 5 and a common difference of 3?

Accepted Answer

5 + 3n

Answer

5 - 3n

Answer

3 + 5n

Answer

3 - 5n

Question 20

Which option represents the sum of the first n terms of an arithmetic sequence with a first term a and a common difference d?

Accepted Answer

n(a + (n-1)d)/2

Answer

n(2a + (n-1)d)/2

Answer

(n+1)(a + (n-1)d)/2

Answer

(n-1)(a + (n+1)d)/2

Question 21

Which formula represents the sum of the first n terms of a geometric sequence with a first term a and a common ratio of r?

Accepted Answer

n(a - r)/(1 - r)

Answer

n(a + r)/(1 - r)

Answer

n(a + r)/(1 + r)

Answer

n(a - r)/(1 + r)

Question 22

Evaluate the limit of the sequence un = (1 + 1/n)^n as n approaches infinity.

Accepted Answer

e

Answer

0

Answer

1

Answer

Infinity

Question 23

Find the nth term of the arithmetic sequence 2, 5, 8, 11, ...

Accepted Answer

2 + 3(n - 1)

Answer

5 + 3(n - 1)

Answer

8 + 3(n - 1)

Answer

11 + 3(n - 1)

Question 24

Which characteristic is NOT applicable to arithmetic sequences?

Accepted Answer

Constant ratio between consecutive terms

Answer

Constant difference between consecutive terms

Answer

Representable by a linear equation

Answer

Common difference

Question 25

Write the equation of the geometric sequence 2, 6, 18, 54, ...

Accepted Answer

a_n = 2(3)^(n-1)

Answer

a_n = 6(3)^(n-1)

Answer

a_n = 18(3)^(n-1)

Answer

a_n = 54(3)^(n-1)

Question 26

Find the nth term of the geometric sequence with first term 3 and common ratio -1/2.

Accepted Answer

3(-1/2)^(n-1)

Answer

3(1/2)^(n-1)

Answer

3(-1/2)^n

Answer

3(1/2)^n

Question 27

In the arithmetic sequence 3, 7, 11, 15, ... what is the constant difference between each term?

Accepted Answer

4

Answer

7

Answer

3

Answer

5

Question 28

Find the nth term of the arithmetic sequence with a first term of 5 and a common difference of 3.

Accepted Answer

5 + 3(n - 1)

Answer

3 + 5(n - 1)

Answer

3 - 5(n - 1)

Answer

5 - 3(n - 1)

Question 29

Find the common ratio of the geometric series 3 + 6 + 12 + 24 + ...

Accepted Answer

2

Answer

1

Answer

4

Answer

3

Question 30

Find the nth term of the arithmetic sequence: 2, 6, 10, 14, ...

Accepted Answer

a_n = 2 + 4(n-1)

Answer

a_n = 2 + 3(n-1)

Answer

a_n = 6 + 4(n-1)

Answer

a_n = 4 + 3(n-1)

Question 31

Determine if the following sequence is arithmetic or geometric: 2, 4, 8, 16, ...

Accepted Answer

Geometric

Answer

Not enough information given

Answer

Neither arithmetic nor geometric

Answer

Arithmetic

Question 32

Find the nth term of the arithmetic sequence whose first term is -7 and common difference is 5.

Accepted Answer

a_n = -7 + 5(n-1)

Answer

a_n = 7 + 5(n-1)

Answer

a_n = -7 + 3(n-1)

Answer

a_n = 7 + 3(n-1)

Question 33

Which of the following is an example of an arithmetic sequence?

Accepted Answer

2, 5, 8, 11, ...

Answer

1, 3, 6, 10, 15, ...

Answer

1, 2, 4, 8, 16, ...

Answer

1, 4, 9, 16, ...

Question 34

Find the common difference of the arithmetic sequence: -3, 0, 3, 6, ...

Accepted Answer

3

Answer

2

Answer

4

Answer

5

Question 35

Find the common ratio of the geometric sequence: 1, 2, 4, 8, ...

Accepted Answer

2

Answer

3

Answer

1

Question 36

Determine the nth term of the arithmetic sequence: 2, 5, 8, 11, ...

Accepted Answer

2 + 3(n-1)

Answer

8 + 3(n-1)

Answer

11 + 3(n-1)

Answer

5 + 2(n-1)

Question 37

Determine the nth term of the geometric sequence: 1, 2, 4, 8, ...

Accepted Answer

1 * 2^(n-1)

Answer

2 * 2^(n-1)

Answer

8 * 2^(n-1)

Answer

4 * 2^(n-1)

Question 38

Find the sum of the first 10 terms of the geometric sequence: 1, 2, 4, 8, ...

Accepted Answer

1023

Answer

512

Answer

2048

Answer

1536

Question 39

Which of the following is NOT a property of geometric sequences?

Accepted Answer

The sum of any two terms is constant.

Answer

The first term is the product of the common ratio and the last term.

Answer

The common ratio is multiplied by each term to obtain the next term.

Answer

The term can be found using the formula: an = a1 * r^(n-1)

Question 40

Consider the arithmetic sequence 2, 5, 8, 11, .... What is the explicit formula for the nth term?

Accepted Answer

3n - 1

Answer

n + 1

Answer

5n - 2

Answer

2n + 3

Question 41

Find the limit of the sequence defined by an = n/(n+1) as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

n

Question 42

Which of the following is the formula for the sum of an infinite geometric series with |r| < 1?

Accepted Answer

a/(1-r)

Answer

a*(1+r)

Answer

a*(1-r)

Answer

a/(1+r)

Question 43

Which of the following is the formula for the nth term of a geometric sequence with first term a and common ratio r?

Accepted Answer

ar^(n-1)

Answer

a * (r^(n-1) - 1)

Answer

a + (n-1) * r

Question 44

Which of the following sequences is an arithmetic sequence, where each term is found by adding a constant value to the previous term?

Accepted Answer

2, 5, 8, 11, ...

Answer

2, 3, 5, 8, ...

Answer

2, 4, 6, 9, ...

Answer

2, 4, 8, 16, ...

Question 45

What is the common difference (the value added to each term) in the arithmetic sequence: 10, 15, 20, 25, ...?

Accepted Answer

5

Answer

20

Answer

15

Answer

10

Question 46

What is the formula for the nth term of the geometric sequence: 2, 6, 18, 54, ...?

Accepted Answer

2 * (3^(n-1))

Answer

6 * (3^(n-1))

Answer

6 * (3^n)

Answer

2 * (3^n)

Question 47

What is the sum of the first 10 terms of the arithmetic sequence: 5, 10, 15, 20, ...?

Accepted Answer

250

Answer

125

Answer

1000

Answer

500

Question 48

What is the sum to infinity of the geometric series: 1 + 1/2 + 1/4 + 1/8 + ...?

Accepted Answer

2

Answer

1/2

Answer

8

Answer

4

Question 49

What is the limit of the sequence: 1/n, as n gets larger and larger (approaches infinity)?

Accepted Answer

0

Answer

∞

Answer

1

Answer

n

Question 50

Which of the following formulas correctly represents the nth term of an arithmetic sequence, where 'a' is the first term and 'd' is the common difference?

Accepted Answer

a + (n - 1) * d

Answer

a + (n + 1) * d

Answer

a - (n - 1) * d

Answer

a - (n + 1) * d

Question 51

If the third term of an arithmetic sequence is 15 and the common difference is 5, what is the first term?

Accepted Answer

5

Answer

10

Answer

15

Answer

20

Question 52

Which of the following formulas correctly represents the sum of the first n terms of an arithmetic sequence, where 'a' is the first term and 'l' is the last term?

Accepted Answer

n/2 * (a + l)

Answer

(a - l) / n

Answer

n/2 * (a - l)

Answer

(a + l) / n

Question 53

If the sum of the first 10 terms of an arithmetic sequence is 150 and the common difference is 3, what is the first term?

Accepted Answer

12

Answer

15

Answer

18

Answer

21

Question 54

Which of the following is the correct formula for the nth term of a geometric sequence?

Accepted Answer

ar^(n-1)

Answer

a^(n+1) / r

Answer

ar^(n+1)

Answer

a^(n-1) / r

Question 55

Which of the following is the correct formula for the sum of the first n terms of a geometric sequence?

Accepted Answer

a(1 - r^n) / (1 - r)

Answer

(1 - r^n) / (1 - r)

Answer

(1 + r^n) / (1 + r)

Answer

a(1 + r^n) / (1 + r)

Question 56

If the sum of the first 6 terms of a geometric sequence is 192 and the common ratio is 2, what is the first term?

Accepted Answer

6

Answer

18

Answer

24

Answer

12