Sequences and Series: Master the Concepts with Interactive Quizzes

Question 1

Which of the following formulas represents the nth term of an arithmetic sequence?

Accepted Answer

a + (n - 1) * d

Answer

a * n + (d - 1)

Answer

a - (n - 1) * d

Answer

a / n + (d - 1)

Question 2

Which of the following expressions accurately represents the sum of the first n terms of an arithmetic sequence?

Accepted Answer

n(a + l) / 2

Answer

(a + l) / 2n

Answer

n(a - l) / 2

Answer

2n(a + l)

Question 3

Which of the following properties is characteristic of a convergent series?

Accepted Answer

The sum of the series approaches a finite value as the number of terms approaches infinity.

Answer

The sum of the series increases without bound as the number of terms approaches infinity.

Answer

The sum of the series alternates between positive and negative values.

Answer

The sum of the series is undefined.

Question 4

What is the main reason for using sigma notation (∑) in the study of sequences and series?

Accepted Answer

To represent the sum of a series

Answer

To find patterns in sequences

Answer

To check if a series converges

Question 5

Which of the following correctly defines the nth term of an arithmetic sequence?

Accepted Answer

an = a1 + (n - 1) * d

Answer

an = a1 + (n + 1) * r

Answer

an = a1 * r^n-1

Answer

an = a1 / r^n-1

Question 6

Determine the common difference of an arithmetic sequence with the first term being 5 and the fifth term being 15.

Accepted Answer

2

Answer

5

Answer

10

Answer

20

Question 7

Identify the formula used to calculate the sum of the first n terms of an arithmetic sequence:

Accepted Answer

Sn = n * (a1 + an) / 2

Answer

Sn = n * (a1 - an) / 2

Answer

Sn = (n - 1) * (a1 + an) / 2

Answer

Sn = (n + 1) * (a1 - an) / 2

Question 8

Which of the following formulas represents the sum to infinity of a convergent geometric series?

Accepted Answer

S = a1 / (1 - r)

Answer

S = a1 * (1 - r)

Answer

S = a1 / (r - 1)

Answer

S = a1 * (r - 1)

Question 9

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

Accepted Answer

The ratio between two consecutive terms is constant.

Answer

The difference between two consecutive terms is constant.

Answer

The first term is given.

Answer

The common difference is given.

Question 10

Which property is NOT characteristic of a convergent series?

Accepted Answer

The terms of the series eventually grow arbitrarily large.

Answer

The sum of a finite number of terms is finite.

Answer

The limit of the sequence of partial sums exists.

Answer

The series can be expressed as the sum of two convergent series.

Question 11

When finding the sum of an alternating series, the most suitable method is:

Accepted Answer

Alternating series test

Answer

Ratio test

Answer

Root test

Answer

Limit comparison test

Question 12

What is the formula for the nth term of an arithmetic sequence with first term a1 and common difference d?

Accepted Answer

an = a1 + (n - 1)d

Answer

an = a1 - (n - 1)d

Answer

an = a1 * (n - 1)d

Answer

an = a1 / (n - 1)d

Question 13

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

Accepted Answer

The ratio of any two consecutive terms is constant.

Answer

The sum of any two terms is equal to the product of the first term and the common ratio.

Answer

The difference between any two consecutive terms is constant.

Answer

The sequence converges to a finite value as n approaches infinity.

Question 14

What is a telescoping series? How can it be used to simplify and evaluate certain infinite series?

Accepted Answer

A telescoping series is a series where consecutive terms cancel out, making evaluation simpler.

Answer

A telescoping series is a sequence where terms approach a finite limit.

Answer

A telescoping series is a series where terms form a geometric progression.

Answer

Telescoping series have a finite number of nonzero terms.

Question 15

Identify the statement that is NOT a characteristic of a geometric sequence:

Accepted Answer

The sequence eventually diverges to infinity.

Answer

The ratio of any two consecutive terms is constant.

Answer

The sum of any two terms exceeds the sum of the two preceding terms.

Answer

Both positive and negative terms can be present within the sequence.

Question 16

Find the 5th term of the sequence defined by the recurrence relation a(n) = 2a(n-1) + 1, where a(1) = 1.

Accepted Answer

13

Answer

5

Answer

33

Answer

65

Question 17

Which application of sequences and series is used in finance?

Accepted Answer

Calculating compound interest

Answer

Modeling population growth

Answer

Predicting weather patterns

Answer

Determining the trajectory of a projectile

Question 18

Write the sigma notation for the sum of the first n odd numbers.

Accepted Answer

Σ(2n-1)

Answer

Σ(n^2)

Answer

Σ(n(n+1))

Answer

Σ(1/n)

Question 19

Find the value of the infinite geometric series: 1 + 1/2 + 1/4 + 1/8 + ...

Accepted Answer

2

Answer

1

Answer

4

Answer

∞

Question 20

Which of the following sequences is a geometric sequence?

Accepted Answer

3, 6, 12, 24, 48

Answer

2, 6, 12, 20, 30

Answer

2, 5, 10, 17, 26

Answer

2, 5, 8, 11, 14

Question 21

A company's profit increases by 15% each year. If the profit in the first year is £12,000, what will be the profit in the fourth year?

Accepted Answer

£17,715

Answer

£16,000

Answer

£17,000

Answer

£18,000

Question 22

A bacteria population triples every hour. If there are initially 500 bacteria, how many bacteria will there be after 5 hours?

Accepted Answer

24,300

Answer

1600

Answer

3200

Answer

12800

Question 23

What is the common difference of the arithmetic sequence -3, -1, 1, 3, ...?

Accepted Answer

2

Answer

3

Answer

4

Answer

1

Question 24

Which of the following is a telescoping series?

Accepted Answer

Sum of 1/(n+1) - 1/n from n=1 to infinity

Answer

Sum of 2^n from n=1 to infinity

Answer

Sum of 1/n from n=1 to infinity

Question 25

What is the common ratio of the geometric sequence 1, 2, 4, 8, ...?

Accepted Answer

2

Answer

4

Answer

1

Question 26

Find the 6th term of the arithmetic sequence whose first term is -5 and the common difference is 3.

Accepted Answer

11

Answer

8

Answer

14

Question 27

Which of the following is the nth term of the geometric sequence 3, 6, 12, ...?

Accepted Answer

3 * 2^(n-1)

Answer

6 * 2^(n-1)

Answer

3 * 2^n

Question 28

What is the sum to infinity of the geometric series 0.5 + 0.25 + 0.125 + ...?

Accepted Answer

1

Answer

0.5

Answer

2

Question 29

Which of the following series converges to a finite value?

Accepted Answer

∑(1/n^2)

Answer

∑(1/n)

Answer

∑(n^2)

Answer

∑(n)

Question 30

Find the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ...

Accepted Answer

2

Answer

Cannot be found since it's not convergent

Answer

∞

Answer

1

Question 31

Which of the following is an application of sequences in the healthcare field?

Accepted Answer

Modeling the spread of infectious diseases

Answer

Solving quadratic equations

Answer

Calculating the volume of a sphere

Answer

Finding the area of a circle

Question 32

Calculate the first 5 terms of the Fibonacci sequence, starting from 1.

Accepted Answer

1, 1, 2, 3, 5

Answer

1, 1, 2, 4, 8

Answer

0, 1, 1, 2, 3

Answer

1, 2, 3, 4, 5

Question 33

Identify the telescoping series among the following.

Accepted Answer

∑(1/n - 1/(n+1))

Answer

∑(n^2)

Answer

∑(1/n^2)

Answer

∑(n)

Question 34

Calculate the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ...

Accepted Answer

2

Answer

1

Answer

1/2

Answer

Cannot be found since it's not convergent

Question 35

Which of the following expressions gives the sum of the first n terms of the arithmetic sequence 5, 10, 15, ...?

Accepted Answer

5n(n + 1)/2

Answer

5n(n - 1)/2

Answer

5n^2(n + 1)

Answer

5n^2(n - 1)

Question 36

Determine the common ratio of the geometric sequence 24, 12, 6, 3, ...

Accepted Answer

1/2

Answer

2

Answer

3

Answer

4

Question 37

What is the common difference of the arithmetic sequence 7, 12, 17, 22, ...?

Accepted Answer

5

Answer

7

Answer

6

Answer

4

Question 38

Which of the following is a geometric sequence?

Accepted Answer

3, 6, 12, 24, ...

Answer

2, 4, 6, 8, ...

Answer

2, 4, 9, 16, ...

Answer

3, 7, 11, 15, ...

Question 39

What is the common ratio of the geometric sequence 1, 2, 4, 8, ...?

Accepted Answer

2

Answer

1

Answer

3

Question 40

What is the common difference of the arithmetic sequence: 3, 7, 11, 15, ...?

Accepted Answer

4

Answer

5

Answer

3

Question 41

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

Accepted Answer

a * r^(n-1)

Answer

r * n - a

Answer

a * n * r

Answer

a + (n-1) * r

Question 42

Find the sum of the first 10 terms of the arithmetic series: 5 + 8 + 11 + 14 + ...

Accepted Answer

125

Answer

200

Answer

150

Answer

175

Question 43

Evaluate the following sigma notation: ∑(n^2 + 1) from n=1 to n=5

Accepted Answer

75

Answer

85

Answer

95

Answer

65

Question 44

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

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 45

Find the nth term of the sequence defined by the recurrence relation: a(n) = 2a(n-1) + 3, where a(1) = 1

Accepted Answer

2^n - 1

Answer

n^2 - 1

Answer

2^n + 1

Question 46

Which of the following applications of sequences and series is used in finance to calculate the future value of an investment?

Accepted Answer

Geometric series

Answer

Pascal's triangle

Answer

Arithmetic sequence

Answer

Fibonacci sequence

Question 47

Find the first four terms of the sequence defined by the explicit formula: a(n) = (-1)^n * 2^n

Accepted Answer

-2, 4, -8, 16

Answer

1, -2, 4, -8

Answer

-1, 2, -4, 8

Answer

2, -4, 8, -16

Question 48

Which of the following is the common ratio of the geometric sequence: 6, 12, 24, 48, ...?

Accepted Answer

2

Answer

3

Answer

1

Answer

4

Question 49

Find the sum of the first 20 terms of the arithmetic series: 10 + 13 + 16 + 19 + ...

Accepted Answer

840

Answer

780

Answer

900

Answer

960