Sequences and Series: Master Arithmetic and Geometric Progressions

Question 1

Which of the following is a correct expansion of (x + y)^3 using the binomial theorem?

Accepted Answer

x^3 + 3x^2y + 3xy^2 + y^3

Answer

x^3 + y^3

Answer

x^2 + 3xy + 3y^2

Answer

x + 3xy + 3y^2 + y

Question 2

Which of the following sequences converges?

Accepted Answer

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

Answer

1, 2, 4, 8, ...

Answer

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

Answer

1, 2, 3, 5, 8, 13, ... (Fibonacci sequence)

Question 3

Find the limit of the sequence 1/n, where n is a natural number.

Accepted Answer

0

Answer

1

Answer

∞

Answer

Does not exist

Question 4

The first four terms of an arithmetic sequence are 3, 8, 13, and 18. What is the common difference?

Accepted Answer

5

Question 5

Find the sum of the first 10 terms of the geometric sequence 2, 4, 8, 16, ...

Accepted Answer

1023

Question 6

Which of the following is the general term of the arithmetic sequence with the first term 5 and common difference -3?

Accepted Answer

a_n = 5 - 3n

Question 7

What is the coefficient of x^3 in the expansion of (2x - 3)^5?

Accepted Answer

240

Question 8

Find the sum of the first 5 terms of the arithmetic series 3 + 7 + 11 + 15 + ...

Accepted Answer

55

Question 9

Which of the following is the general term of the geometric series 2 + 4 + 8 + 16 + ...?

Accepted Answer

a_n = 2 * 2^n

Question 10

Find the 10th term of the arithmetic sequence with the first term 10 and common difference 5.

Accepted Answer

65

Answer

60

Question 11

Which of the following is the formula for the nth term (a_n) of an arithmetic sequence with first term (a_1), common difference (d), and n being the term number?

Accepted Answer

a_n = a_1 + (n - 1)d

Answer

a_n = a_1 * r^(n - 1)

Answer

a_n = a_1 + (n - 1)r

Answer

a_n = a_1 * d^(n - 1)

Question 12

Find the 5th term of the geometric sequence whose first term is 4 and common ratio is 3.

Accepted Answer

243

Answer

216

Answer

270

Answer

297

Question 13

Which of the following is the nth term of the arithmetic sequence whose first term is 5 and common difference is -2?

Accepted Answer

5 - 2(n - 1)

Answer

5 + 2(n - 1)

Answer

5 - 2(n + 1)

Answer

5 + 2(n + 1)

Question 14

What is the value of the following sum: 1 + 3 + 9 + 27 + 81 + ... + 3^10?

Accepted Answer

5184

Answer

1296

Answer

2592

Answer

10368

Question 15

Which of the following is the domain of the sequence defined by g(n) = sqrt(n + 2)?

Accepted Answer

{n ∈ R | n ≥ -2}

Answer

{n ∈ N}

Answer

{n ∈ Z | n ≥ -2}

Answer

{n ∈ Q | n ≥ -2}

Question 16

Identify the characteristic that is NOT applicable to a geometric sequence:

Accepted Answer

Constant term

Answer

Common ratio

Answer

All terms are non-zero

Answer

Sum of the first n terms forms a geometric series

Question 17

What is the formula for the sum of the first n terms of an arithmetic sequence?

Accepted Answer

n(a + l)/2

Answer

n(a - l)/2

Answer

n(2a + l)/2

Answer

n(a + 2l)/2

Question 18

Given the sequence defined by a(n) = 2n^2 + 3n - 1, calculate the difference between the 5th and 3rd terms.

Accepted Answer

28

Answer

20

Answer

36

Answer

40

Question 19

The first term of a geometric sequence is 5 and the common ratio is 2. What is the 5th term of the sequence?

Accepted Answer

160

Answer

40

Answer

80

Answer

320

Question 20

Which of the following is the middle term of the arithmetic progression 5, 9, 13, 17, 21?

Accepted Answer

13

Answer

9

Answer

17

Answer

21

Question 21

Which of the following is the general term of the geometric series 2 + 4 + 8 + 16 + ...?

Accepted Answer

2^n

Answer

2^(n-1)

Answer

2^(n+1)

Answer

2^(n+2)

Question 22

Calculate the sum of the first n terms of the geometric sequence {2, 6, 18, 54, ...}.

Accepted Answer

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

Answer

2(3^n - 1)

Answer

3(2^n - 1)

Answer

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

Question 23

Determine the nth term of the sequence {1, 4, 9, 16, 25, ...}.

Accepted Answer

n^2

Answer

n^2 - 1

Answer

2n^2 - 1

Answer

2n^2

Question 24

Calculate the sum of the first 50 terms of the arithmetic sequence {2, 5, 8, 11, ...}.

Accepted Answer

1250

Answer

1000

Answer

1500

Answer

1750

Question 25

What is the general term of an arithmetic sequence with a first term of 5 and a common difference of 3?

Accepted Answer

5 + 3n

Answer

3 + 5n

Answer

3n + 5

Answer

5n + 3

Question 26

Which of the following is the nth term of the arithmetic sequence 5, 9, 13, ...?

Accepted Answer

4(n - 1) + 5

Answer

4n + 5

Answer

4(n - 1) + 9

Answer

4n + 9

Question 27

Write the Binomial Theorem expansion for (x - 2)^3.

Accepted Answer

x^3 - 6x^2 + 12x - 8

Answer

x^3 + 6x^2 - 12x - 8

Answer

x^3 - 6x^2 + 12x + 8

Answer

x^3 + 6x^2 + 12x + 8

Question 28

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

Accepted Answer

The ratio of any two consecutive terms is constant.

Answer

Each term is multiplied by a constant.

Answer

The difference between any two consecutive terms is constant.

Answer

The terms increase linearly.

Question 29

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

Accepted Answer

Sn = n * (a1 - r) / (1 - r)

Answer

Sn = n * (a1 + r) / (1 + r)

Answer

Sn = n * (a1 - r) / (1 + r)

Answer

Sn = n * (a1 + r) / (1 - r)

Question 30

Find the nth term of the following arithmetic sequence: 10, 13, 16, 19, ...

Accepted Answer

a_n = 10 + 3 * (n-1)

Answer

a_n = 10 + 3 * n

Answer

a_n = 10 + (n-1)

Answer

a_n = 10 + n

Question 31

Which of the following sequences is an arithmetic sequence, where each term after the first is obtained by adding a constant difference?

Accepted Answer

1, 3, 5, 7, 9

Answer

2, 4, 8, 16, 32

Answer

1, 2, 4, 8, 16

Answer

1, 1/2, 1/4, 1/8, 1/16

Question 32

Which of the following sequences is divergent, meaning its terms do not approach a single finite value?

Accepted Answer

1, 2, 4, 8, ...

Answer

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

Answer

1, 3, 5, 7, ...

Answer

1, 1/3, 1/9, 1/27, ...

Question 33

Find the Maclaurin series representation for the function f(x) = e^x, which expresses the function as an infinite sum of terms involving x and its derivatives evaluated at x=0

Accepted Answer

1 + x + x^2/2! + x^3/3! + ...

Answer

1 - x + x^2/2! - x^3/3! + ...

Answer

1 + x - x^2/2! + x^3/3! + ...

Answer

1 - x - x^2/2! - x^3/3! + ...

Question 34

Which expression represents the binomial expansion of (x + 2)^3?

Accepted Answer

x^3 + 6x^2 + 12x + 8

Answer

x^3 + 8x^2 + 12x + 16

Answer

x^3 + 6x^2 + 18x + 16

Answer

x^3 + 8x^2 + 18x + 32

Question 35

Identify the necessary condition for a sequence to be arithmetic.

Accepted Answer

The common difference between any two consecutive terms is constant.

Answer

The sum of any two consecutive terms is constant.

Answer

The product of any two consecutive terms is constant.

Answer

The quotient of any two consecutive terms is constant.

Question 36

Which condition must be met for a sequence to be geometric?

Accepted Answer

The common ratio between any two consecutive terms is constant.

Answer

The sum of any two consecutive terms is constant.

Answer

The product of any two consecutive terms is constant.

Answer

The difference between any two consecutive terms is constant.

Question 37

Identify the common ratio of the geometric sequence 2, 6, 18, 54, ...

Accepted Answer

3

Answer

4

Answer

2

Answer

6

Question 38

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

Accepted Answer

2

Answer

3

Answer

Converges to infinity

Answer

4

Question 39

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

Accepted Answer

The sum of any two consecutive terms is constant.

Answer

The sequence can be extended indefinitely.

Answer

The ratio of any two consecutive terms is constant.

Answer

The common ratio is greater than 0.

Question 40

Expand (x + 2)⁴ using the Binomial Theorem. Which of the following is the correct expansion?

Accepted Answer

x⁴ + 8x³ + 24x² + 32x + 16

Answer

x⁴ + 16x³ + 24x² + 16x + 1

Answer

x⁴ + 8x³ + 16x² + 16x + 4

Answer

x⁴ + 4x³ + 6x² + 4x + 1

Question 41

What is the common ratio of the geometric series: 1/2, 1, 2, 4, ...?

Accepted Answer

2

Answer

1/2

Answer

1

Answer

4

Question 42

The geometric sequence 3, 6, 12, 24, ... has a common ratio of:

Accepted Answer

2

Answer

6

Answer

4

Answer

3

Question 43

The first term of an arithmetic sequence is 5 and the common difference is 3. What is the sum of the first 10 terms of the sequence?

Accepted Answer

195

Answer

175

Answer

215

Answer

205

Question 44

The sum of the infinite geometric series 1 + 1/2 + 1/4 + ... is:

Accepted Answer

2

Answer

5/4

Answer

3/2

Answer

1

Question 45

The sum of the first n natural numbers is given by the formula:

Accepted Answer

n(n+1)/2

Answer

n^2

Answer

n(n-1)/2

Answer

n(n+2)/2

Question 46

The coefficient of x^2 in the expansion of (x + 2)^4 is:

Accepted Answer

24

Answer

36

Answer

48

Answer

12

Question 47

What is the 10th term of the arithmetic sequence: 3, 7, 11, 15... ?

Accepted Answer

39

Answer

43

Answer

47

Answer

35

Question 48

The first term of a geometric sequence is 2 and the common ratio is 3. What is the 5th term?

Accepted Answer

162

Answer

486

Answer

54

Answer

1458

Question 49

Find the sum of the first 20 terms of the arithmetic series: 5 + 8 + 11 + ...

Accepted Answer

670

Answer

610

Answer

730

Question 50

The first term of a geometric series is 4, and the common ratio is 0.5.  What is the sum to infinity of the series?

Accepted Answer

8

Answer

4

Answer

The series does not converge

Answer

16

Question 51

What is the coefficient of the x³ term in the expansion of (2x - 1)⁵?

Accepted Answer

-80

Answer

80

Answer

-40

Answer

40

Question 52

An arithmetic sequence has a 5th term of 17 and a 10th term of 32. What is the common difference?

Accepted Answer

3

Answer

4

Answer

2

Question 53

A geometric sequence has a second term of 6 and a fifth term of 162. What is the common ratio?

Accepted Answer

3

Answer

2

Answer

-3

Question 54

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 does not converge

Question 55

What is the coefficient of the x² term in the expansion of (x + 2)⁴?

Accepted Answer

24

Answer

48

Answer

6

Answer

12

Question 56

The sum of the first 15 terms of an arithmetic series is 360.  If the first term is 12, what is the common difference?

Accepted Answer

4

Answer

3

Answer

2

Answer

6