Sequences and Series: Master Arithmetic and Geometric Progressions

Question 1

Which of the following is the formula for the nth term of an arithmetic sequence?

Accepted Answer

a + (n - 1)d

Answer

a + n^2d

Answer

nd + a

Answer

a^nd

Question 2

Identify the geometric sequence from the following options:

Accepted Answer

2, 6, 18, 54, 162

Answer

2, 4, 6, 8, 10

Answer

1, 3, 5, 7, 9

Answer

1, 2, 3, 4, 5

Question 3

A store offers a 10% discount on all purchases over $50. If a customer spends $75, how much will they save?

Accepted Answer

$7.50

Answer

$5

Answer

$10

Answer

$12.50

Question 4

A population of bacteria doubles every hour. If there are initially 100 bacteria, how many bacteria will there be after 5 hours?

Accepted Answer

1600

Answer

200

Answer

400

Answer

800

Question 5

Which of the following is the formula for the sum of an infinite geometric series with a common ratio less than 1?

Accepted Answer

S = a / (1 - r)

Answer

S = a * (1 - r)

Answer

S = a / r

Answer

S = a * r

Question 6

Which of the following sequences is convergent?

Accepted Answer

Un = 1/n

Answer

Un = n

Answer

Un = n^2

Answer

Un = (-1)^n

Question 7

Determine the common difference of the arithmetic progression 5, 11, 17, ...

Accepted Answer

3

Answer

2

Answer

4

Answer

5

Question 8

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

Accepted Answer

Sn = n(U1 + Un) / 2

Answer

Sn = n(U1 - Un) / 2

Answer

Sn = n * U1

Answer

Sn = n * Un

Question 9

Identify the common ratio of the geometric progression 1, 2, 4, ...

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 10

Which formula represents the sum of the first n terms of a geometric series?

Accepted Answer

Sn = U1(1 - r^n) / (1 - r)

Answer

Sn = U1(1 + r^n) / (1 + r)

Answer

Sn = U1 * r^n

Answer

Sn = U1 * n

Question 11

An arithmetic sequence has its first term as 5 and a common difference of 3. Find the 10th term of this sequence.

Accepted Answer

32

Answer

15

Answer

28

Question 12

Identify the general term of the sequence: 2, 4, 8, 16, ...

Accepted Answer

2^n

Answer

2^(n-1)

Answer

2n

Answer

2(n-1)

Question 13

Determine the common difference of the arithmetic sequence: -5, -2, 1, 4, ...

Accepted Answer

3

Answer

5

Answer

7

Answer

9

Question 14

Which of the following expressions represents the nth term of the geometric sequence 3, 9, 27, 81, ...?

Accepted Answer

3^n

Answer

9^n

Answer

3^(n-1)

Answer

9^(n-1)

Question 15

Given the geometric series 1, 1/2, 1/4, 1/8, ... , what is the sum to infinity of this series?

Accepted Answer

2

Answer

1

Answer

4

Answer

Cannot be determined

Question 16

If the 10th term of an arithmetic sequence is 45 and the common difference is 3, what is the first term of the sequence?

Accepted Answer

21

Answer

15

Answer

27

Answer

33

Question 17

Which of the following is a property of geometric sequences?

Accepted Answer

The ratio of consecutive terms is constant.

Answer

The sum of any two terms is equal to the common ratio.

Answer

The first term is always equal to the common ratio.

Answer

The sequence can only contain positive terms.

Question 18

Find the common ratio of the geometric sequence: 12, 6, 3, 1.5, ...

Accepted Answer

0.5

Answer

1

Answer

2

Answer

4

Question 19

Which of the following series converges?

Accepted Answer

∑(n=1 to ∞) 1/n^2

Answer

∑(n=1 to ∞) 1/n

Answer

∑(n=1 to ∞) n

Answer

∑(n=1 to ∞) n^2

Question 20

Which of the following formulas represents 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 21

Which of the following sequences is not an arithmetic sequence?

Accepted Answer

1, 4, 9, 16, ...

Answer

3, 5, 7, 9, ...

Answer

2, 6, 10, 14, ...

Answer

3, 6, 12, 24, ...

Question 22

Which of the following functions represents a geometric sequence?

Accepted Answer

f(n) = 2^n

Answer

f(n) = 3n + 1

Answer

f(n) = n^2 - 2

Answer

f(n) = 1 / n^2

Question 23

The nth partial sum of the series 2 + 4 + 8 + 16 + ... is given by:

Accepted Answer

Sn = 2 * (1 - 2^n)

Answer

Sn = 2 * (1 + 2^n)

Answer

Sn = 4 * (1 - 2^n)

Answer

Sn = 4 * (1 + 2^n)

Question 24

In an arithmetic sequence, the difference between any two consecutive terms is:

Accepted Answer

Always equal

Answer

Always different

Answer

May be equal or different

Answer

Zero

Question 25

The nth term of an arithmetic sequence is given by:

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 26

The sum of the first n terms of an arithmetic sequence is given by:

Accepted Answer

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

Answer

Sn = n/2 * (a1 + an)

Answer

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

Answer

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

Question 27

The nth term of a geometric sequence is given by:

Accepted Answer

an = a1 * r^(n-1)

Answer

an = a1 * r^(n)

Answer

an = a1 / r^(n-1)

Answer

an = a1 / r^(n)

Question 28

The sum of an infinite geometric series with a common ratio less than 1 and a first term of a is:

Accepted Answer

a / (1 - r)

Answer

a / (1 + r)

Answer

a * (1 - r)

Answer

a * (1 + r)

Question 29

Which of the following is NOT a valid formula for the common ratio of a geometric sequence?

Accepted Answer

r = a1 / an

Answer

r = an / an-1

Answer

r = an-1 / an

Answer

r = an / a1

Question 30

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

Accepted Answer

34

Answer

26

Answer

30

Answer

38

Question 31

Given the sequence: 5, 10, 15, 20, ... find the common difference.

Accepted Answer

5

Answer

10

Answer

15

Answer

20

Question 32

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

Accepted Answer

Constant difference

Answer

Common ratio

Answer

Multiplication pattern

Answer

Convergent nature

Question 33

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

Accepted Answer

The first term is zero.

Answer

Each term is obtained by multiplying the previous term by a constant.

Answer

The common ratio is the same for all terms.

Answer

The sum of any two terms is equal to the next term.

Question 34

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

Accepted Answer

2

Answer

1

Answer

4

Answer

8

Question 35

Which of the following is a convergent geometric series?

Accepted Answer

1 + 1/2 + 1/4 + 1/8 + ...

Answer

1 - 1/2 + 1/4 - 1/8 + ...

Answer

1 + 2 + 4 + 8 + ...

Answer

1 + 1/2 + 1/3 + 1/4 + ...

Question 36

Which of the following is the formula for the nth term of an arithmetic sequence?

Accepted Answer

a_n = a_1 + (n - 1) * d

Answer

a_n = a_1 * (n - 1) * d

Answer

a_n = a_1 + (n - 1) * r

Answer

a_n = a_1 * (n - 1) * r

Question 37

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

Accepted Answer

a_n = a_1 * (n - 1) * r

Answer

a_n = a_1 + (n - 1) * d

Answer

a_n = a_1 * (n - 1) * d

Answer

a_n = a_1 + (n - 1) * r

Question 38

Which of the following sequences is geometric?

Accepted Answer

{2, 4, 8, 16, 32}

Answer

{1, 3, 5, 7, 9}

Answer

{1, 2, 4, 8, 16}

Answer

{1, 4, 9, 16, 25}

Question 39

What is the common ratio of the geometric sequence {2, 4, 8, 16, 32}?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 40

A bacterial population doubles every hour. If there are initially 100 bacteria, how many will there be after 6 hours?

Accepted Answer

12800

Answer

1600

Answer

3200

Answer

6400

Question 41

Find the common ratio of the geometric sequence: 6, 12, 24, 48, ...

Accepted Answer

2

Answer

1/2

Answer

4

Answer

8

Question 42

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

Accepted Answer

3

Answer

1

Answer

5

Answer

7

Question 43

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

Accepted Answer

2

Answer

1

Answer

4

Answer

The series diverges

Question 44

A population of bacteria doubles every hour. If the initial population is 100, what will be the population after 6 hours?

Accepted Answer

1000

Answer

400

Answer

600

Answer

800

Question 45

Given the sequence {1, 1/2, 1/4, 1/8, ...}, what is its limit?

Accepted Answer

0

Answer

1

Answer

1/2

Answer

Does not exist

Question 46

Find the 10th term of the arithmetic progression with a first term of 5 and a common difference of 3.

Accepted Answer

25

Answer

22

Answer

28

Answer

31

Question 47

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

Accepted Answer

a * r^(n - 1)

Answer

a + (n - 1) * r

Answer

ar^n

Answer

r * a^(n - 1)

Question 48

A company's profit doubles every year. If the profit in the first year was $10,000, what will be the profit in the 5th year?

Accepted Answer

$640,000

Answer

$80,000

Answer

$160,000

Answer

$320,000

Question 49

Which of the following sequences is a Fibonacci sequence?

Accepted Answer

1, 1, 2, 3, 5

Answer

1, 2, 3, 4, 5

Answer

1, 2, 4, 8, 16

Answer

1, 3, 6, 10, 15

Question 50

Find the common ratio of the geometric sequence: 1/3, 1, 3, 9, ...

Accepted Answer

3

Answer

1/3

Answer

9

Answer

27

Question 51

Determine the nth term of the arithmetic sequence with a(1) = 5 and common difference d = 3

Accepted Answer

5 + 3n

Answer

3n + 2

Answer

5 - 3n

Answer

5n + 3

Question 52

Determine the first term of the geometric sequence with a(5) = 243 and r = 3

Accepted Answer

3

Answer

1

Answer

9

Answer

27

Question 53

Find the first five terms of the sequence defined recursively by a(0) = 2 and a(n) = 2a(n-1) + 1.

Accepted Answer

2, 5, 11, 23, 47

Answer

2, 4, 9, 19, 39

Answer

2, 3, 7, 15, 31

Answer

2, 5, 10, 20, 40

Question 54

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

Accepted Answer

The product of any two terms is equal to the square of the middle term.

Answer

The difference between any two consecutive terms is constant.

Answer

The sum of any two terms is equal to the middle term.

Answer

The sequence can be represented by a linear equation.