Sequences and Series: Master Patterns and Convergence in Math

Question 1

Which of the following statements about the sequence {n^2} is true?

Accepted Answer

It is both increasing and unbounded.

Answer

It is increasing but bounded.

Answer

It is decreasing and unbounded.

Answer

It is decreasing but bounded.

Question 2

Which of the following is NOT a type of sequence?

Accepted Answer

Binomial sequence

Answer

Arithmetic sequence

Answer

Geometric sequence

Answer

Harmonic sequence

Question 3

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) * d

Answer

a[n] = a[1] - (n+1) * d

Question 4

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

Accepted Answer

a[n] = a[1] * (r^(n-1))

Answer

a[n] = a[1] * (r^(n+1))

Answer

a[n] = a[1] / (r^(n-1))

Answer

a[n] = a[1] / (r^(n+1))

Question 5

Determine whether the following sequence is convergent or divergent: 1/2, 1/4, 1/8, 1/16, ...

Accepted Answer

Converges

Answer

Diverges

Question 6

Which of the following is not a recognized type of sequence?

Accepted Answer

Quadratic

Answer

Arithmetic

Answer

Geometric

Answer

Harmonic

Question 7

Determine whether the following sequence is monotonically increasing or monotonically decreasing: 10, 8, 6, 4, 2

Accepted Answer

Monotonically decreasing

Answer

Monotonically increasing

Answer

Neither increasing nor decreasing

Question 8

What is the limit of the sequence 1/2, 1/4, 1/8, 1/16, ...?

Accepted Answer

0

Answer

1/2

Answer

1

Answer

2

Question 9

To find the sum of the first n terms of an arithmetic sequence, which formula should you use?

Accepted Answer

Sn = n(a1 + an) / 2

Answer

Sn = n(a1 - an) / 2

Answer

Sn = a1 + an / 2

Answer

Sn = (a1 + an) / 2

Question 10

Determine whether the following series converges or diverges: 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

Accepted Answer

Converges

Answer

Diverges

Question 11

Find the nth term of the arithmetic sequence with first term 7 and common difference 5.

Accepted Answer

7 + 5n

Answer

7 - 5n

Answer

5n + 7

Answer

5n - 7

Question 12

Find the explicit formula for the sequence {1, 3, 5, 7, ...}.

Accepted Answer

a_n = 2n - 1

Answer

a_n = n

Answer

a_n = n + 1

Answer

a_n = 2n + 1

Question 13

Which of the following sequences is geometric?

Accepted Answer

{2, 4, 8, 16, 32}

Answer

{1, 2, 3, 4, 5}

Answer

{1, 3, 5, 7, 9}

Answer

{2, 3, 5, 8, 12}

Question 14

Find the common difference of the arithmetic sequence {5, 10, 15, 20, ...}.

Accepted Answer

3

Answer

5

Answer

7

Answer

9

Question 15

Which term describes a sequence with a consistent difference between any two consecutive terms?

Accepted Answer

Arithmetic

Answer

Geometric

Answer

Fibonacci

Answer

Harmonic

Question 16

For the geometric sequence {2, 6, 18, 54, ...}, what is the nth term?

Accepted Answer

2 * 3^(n-1)

Answer

3 * 2^(n-1)

Answer

2 * 2^(n+1)

Answer

3 * 3^(n-1)

Question 17

Find the limit of the sequence {1, 1/2, 1/4, 1/8, ...} as n approaches infinity.

Accepted Answer

0

Answer

1/2

Answer

1

Answer

It diverges

Question 18

Find the sum of the first 10 terms of the arithmetic series 2 + 4 + 6 + ...

Accepted Answer

100

Answer

50

Answer

150

Answer

200

Question 19

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

Accepted Answer

a + (n - 1) * d

Answer

a * (n - 1) * d

Answer

a / (n - 1) * d

Answer

a - (n - 1) * d

Question 20

Consider the series: 3 + 6 + 9 + 12 + ... Find the sum of the first 10 terms of this arithmetic series.

Accepted Answer

230

Answer

220

Answer

240

Answer

250

Question 21

Find the common ratio of the sequence: 2, 6, 18, 54, ...

Accepted Answer

3

Answer

2

Answer

4

Answer

5

Question 22

Which of the following sequences converges to a finite limit?

Accepted Answer

{n / 2^n}

Answer

{n^2 / n + 1}

Answer

{(-1)^n}

Answer

{n!}

Question 23

Which of the following series is absolutely convergent?

Accepted Answer

∑(n=1 to ∞) 1/2^n

Answer

∑(n=1 to ∞) 1/n

Answer

∑(n=1 to ∞) n

Question 24

Determine the interval of convergence for the following series: ∑(n=1 to ∞) (x^n)/n

Accepted Answer

-1 < x < 1

Answer

x = 0

Answer

x < -1

Answer

x > 1

Question 25

Which of the following accurately represents the formula for the nth term (an) of an arithmetic sequence with the first term (a) and common difference (d)?

Accepted Answer

an = a + (n - 1) * d

Answer

an = a - (n - 1) * d

Answer

an = (n - 1) * a + d

Answer

an = d + (n - 1) * a

Question 26

Determine whether the following series is convergent or divergent: 1 + 1 / 2 + 1 / 4 + 1 / 8 + ...

Accepted Answer

Convergent

Answer

None of the above

Answer

Cannot be determined

Answer

Divergent

Question 27

Which test is specifically designed to determine the convergence or divergence of an alternating series?

Accepted Answer

Alternating Series Test

Answer

Comparison Test

Answer

Ratio Test

Answer

Limit Comparison Test

Question 28

Which formula accurately represents the nth term (an) of a geometric sequence, given the first term (a) and common ratio (r)?

Accepted Answer

an = a * r^(n - 1)

Answer

an = a + (n - 1) * r

Answer

an = r^(n - 1) + a

Answer

an = (n - 1) * a * r

Question 29

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

Accepted Answer

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

Answer

a + r^n

Answer

n * (a + r^n)

Answer

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

Question 30

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

Accepted Answer

a + (n - 1)d

Answer

a + (n + 1)d

Answer

a - (n + 1)d

Answer

a - (n - 1)d

Question 31

Determine the common difference of the arithmetic sequence: 4, 7, 10, 13, ...

Accepted Answer

3

Answer

5

Answer

4

Answer

2

Question 32

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

Accepted Answer

-1/2

Answer

1/2

Answer

2

Answer

-2

Question 33

Find the sum of the convergent series: 2 + 1 + 1/2 + 1/4 + ...

Accepted Answer

4

Answer

8

Answer

6

Answer

2

Question 34

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

Accepted Answer

n/2 * (a1 + an)

Answer

n/2 * (2a1 - an)

Answer

n/2 * (2a1 + an)

Answer

n/2 * (a1 - an)

Question 35

Find the sum of the first 12 terms of the arithmetic sequence: 6, 11, 16, 21, ...

Accepted Answer

468

Answer

568

Answer

368

Answer

268

Question 36

Which of the following sequences is a decreasing sequence?

Accepted Answer

-2, -4, -6, -8, ...

Answer

1, 2, 3, 4, ...

Answer

0, 1, 2, 3, ...

Answer

-1, 0, 1, 2, ...

Question 37

Find the limit of the sequence: (3n^2 + 2n - 1) / (n^2 + n + 1)

Accepted Answer

3

Answer

1

Answer

2

Question 38

What is the sum of the series 1 + 2 + 4 + 8 + ...?

Accepted Answer

Diverges

Answer

4

Answer

16

Answer

8

Question 39

What is the nth term of the sequence defined by a_n = 2n - 5?

Accepted Answer

2n - 5

Answer

n - 5

Answer

2n + 5

Answer

5 - 2n

Question 40

Which of the following is an example of a telescoping series?

Accepted Answer

1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... (terms cancel out in pairs, leaving only the first and last terms)

Answer

1 + 2 + 4 + 8 + ... (geometric series)

Answer

1 + 1/2 + 1/4 + 1/8 + ... (geometric series)

Answer

1 + 1/2 + 1/3 + 1/4 + ... + 1/n (harmonic series)

Question 41

The harmonic series is the series 1/2 + 1/4 + 1/8 + ... . What is its sum?

Accepted Answer

Diverges (the sum of the series grows without bound)

Answer

2 (the sum of the first two terms)

Answer

1 (the sum of the first term)

Answer

Infinity (the sum of an infinite number of terms)

Question 42

Find the next term in the arithmetic sequence: 2, 5, 8, 11, ...

Accepted Answer

14 (the common difference is 3)

Answer

13 (incorrect; the common difference is 3)

Answer

15 (incorrect; the common difference is 3)

Answer

12 (incorrect; the common difference is 3)

Question 43

Find the 10th term of the geometric sequence: 3, 6, 12, 24, ...

Accepted Answer

1536 (the common ratio is 2)

Answer

6144 (incorrect; the common ratio is 2)

Answer

3072 (incorrect; the common ratio is 2)

Answer

768 (incorrect; the common ratio is 2)

Question 44

Which of the following series is convergent (has a finite sum)?

Accepted Answer

1 + 1/2 + 1/4 + 1/8 + ... (geometric series with common ratio 1/2)

Answer

1 + 2 + 4 + 8 + ... (geometric series with common ratio 2)

Answer

1 + 1/3 + 1/9 + 1/27 + ... (geometric series with common ratio 1/3)

Answer

1 - 1 + 1 - 1 + ... (alternating series)

Question 45

Calculate the sum of the infinite geometric series: 1 + 1/3 + 1/9 + 1/27 + ...

Accepted Answer

3/2 (the common ratio is 1/3)

Answer

1 (incorrect; the sum of the first term only)

Answer

2 (incorrect; incorrect calculation)

Question 46

Find the sum of the first 10 terms of the arithmetic series: 2 + 5 + 8 + ...

Accepted Answer

55 (the common difference is 3, and the sum of the first n terms of an arithmetic series is n/2 * (2a + (n-1)d))

Answer

65 (incorrect; incorrect calculation)

Answer

45 (incorrect; incorrect calculation)

Answer

35 (incorrect; incorrect calculation)

Question 47

The sequence defined by the formula a_n = 2n - 1 is:

Accepted Answer

Arithmetic (each term is obtained by adding 2 to the previous term)

Answer

Geometric (each term is obtained by multiplying the previous term by a constant ratio)

Answer

Harmonic (each term is the reciprocal of the previous term)

Answer

Fibonacci (each term is the sum of the two preceding terms)

Question 48

A sequence where each term is the sum of the two preceding terms is called a:

Accepted Answer

Fibonacci Sequence

Answer

Arithmetic Sequence

Answer

Geometric Sequence

Question 49

A series is said to converge if:

Accepted Answer

The sum of its terms approaches a finite value.

Answer

The sum of its terms approaches infinity.

Answer

The terms of the series increase without bound.

Answer

The terms of the series alternate between positive and negative values.

Question 50

The formula for the sum of a finite arithmetic series is:

Accepted Answer

S_n = (n/2)(a_1 + a_n)

Answer

S_n = n(a_1 * a_n)

Answer

S_n = (n/2)(a_1 * a_n)

Answer

S_n = n(a_1 + a_n)