Free Pre-Calculus Sequences and Series Questions and Answers — Questions and Answers

Question 1: Create the geometric sequence's formula: -2, 10, -50, 250...

aₙ = 2(-5)^n-1
aₙ = -2(5)^n-1
aₙ = 2(5)^n-1
aₙ = -2(-5)^n-1 (Correct answer)

Correct answer: aₙ = -2(-5)^n-1

A geometric sequence has a first term (a₁) and a common ratio (r), with the formula aₙ = a₁ * r^(n-1). From the sequence -2, 10, -50, 250..., the first term a₁ is -2. The common ratio r is found by dividing any term by its preceding term: 10 / (-2) = -5. Substituting these values into the formula gives aₙ = -2(-5)^(n-1).

Question 2: Eliya's Bakery has recently launched and is expanding its bread products. For instance, the bakery produced 150 carrot cakes in January, 133 in February, 116 in March, and 99 in April. What type of sequence is this?

Arithmetic (Correct answer)
Geometric
Neither
Both

Correct answer: Arithmetic

To classify the sequence, we check for a common difference (arithmetic) or a common ratio (geometric). The differences between consecutive terms are: 133 - 150 = -17, 116 - 133 = -17, and 99 - 116 = -17. Since there is a constant difference of -17, the sequence is arithmetic. It is not geometric as the ratios between terms are not constant.

Question 3: Determine the sum, if it exists, of the infinite geometric series: 200 - 100 + 50 - 25 +

3 / 400
2 / 400
Does not exist
400/ 3 (Correct answer)

Correct answer: 400/ 3

This is an infinite geometric series with a first term (a) of 200 and a common ratio (r) of -100/200 = -1/2. Since the absolute value of the common ratio (|r| = 1/2) is less than 1, the sum exists. Using the formula S = a / (1 - r), we get S = 200 / (1 - (-1/2)) = 200 / (3/2) = 400/3.

Question 4: A geometric sequence's third term is 96 and its fifth term is 1536. What is the sum of the sequence's first ten terms?

33,554,400
2,097,150 (Correct answer)
4,092
1,572,864

Correct answer: 2,097,150

First, find the common ratio (r) using the terms a_5 = a_3 * r^2, which gives 1536 = 96 * r^2, so r^2 = 16, and r = 4 (since terms are positive). Then, find the first term (a_1) using a_3 = a_1 * r^2, so 96 = a_1 * 4^2, which means a_1 = 6. Finally, use the sum formula for a geometric series S_n = a_1 * (1 - r^n) / (1 - r) to find S_10 = 6 * (1 - 4^10) / (1 - 4) = 2,097,150.

Question 5: Determine the common difference 97, 86, 75, 64, ...

11
-8
-11 (Correct answer)
8

Correct answer: -11

The common difference in an arithmetic sequence is found by subtracting any term from its succeeding term. For this sequence, 86 - 97 = -11. We can verify this by checking other consecutive terms, such as 75 - 86 = -11 and 64 - 75 = -11, confirming the common difference is -11.

Question 6: Determine the 22nd term of the sequence 5, 8, 11,...

63
71
68 (Correct answer)
14

Correct answer: 68

This is an arithmetic sequence with a first term (a_1) of 5 and a common difference (d) of 8 - 5 = 3. To find the 22nd term (a_22), use the formula a_n = a_1 + (n-1)d. Plugging in the values, a_22 = 5 + (22-1) * 3 = 5 + 21 * 3 = 5 + 63 = 68.

Question 7: Determine the sum of the terms in the sequence 1, 1/2, 1/4, 1/8, and 1/16...

3
1
4
2 (Correct answer)

Correct answer: 2

This is an infinite geometric series where the first term (a) is 1 and the common ratio (r) is (1/2) / 1 = 1/2. Since the absolute value of the common ratio (|r| = 1/2) is less than 1, the sum converges. Using the formula for the sum of an infinite geometric series, S = a / (1 - r), we get S = 1 / (1 - 1/2) = 1 / (1/2) = 2.

Question 8: Your trainer instructs you to return to your jogging program gradually following leg surgery. He recommends 12 minutes of daily jogging for the first week. Each subsequent week, he suggests increasing this time by 6 minutes per day. In how many weeks will you be able to jog for 60 minutes per day?

10 weeks
1 weeks
9 weeks (Correct answer)
8 weeks

Correct answer: 9 weeks

This scenario represents an arithmetic sequence where the first term (a_1) is 12 minutes and the common difference (d) is 6 minutes per week. We want to find the week (n) when the jogging time (a_n) reaches 60 minutes. Using the formula a_n = a_1 + (n-1)d, we have 60 = 12 + (n-1)6. Solving for n gives 48 = (n-1)6, so 8 = n-1, which means n = 9 weeks.

Question 9: When you go to the Grand Canyon, you throw a penny off a cliff. In arithmetic order, the penny will fall 16 feet in the first second, 48 feet in the second after that, 80 feet in the third second, and so on. How far will the object fall in total in six seconds?

480 feet
576 feet (Correct answer)
120 feet
144 feet

Correct answer: 576 feet

This is an arithmetic series where the first term (a_1) is 16 feet and the common difference (d) is 48 - 16 = 32 feet. To find the total distance fallen in six seconds (S_6), use the sum formula S_n = n/2 * (2a_1 + (n-1)d). Plugging in the values, S_6 = 6/2 * (2*16 + (6-1)*32) = 3 * (32 + 5*32) = 3 * (32 + 160) = 3 * 192 = 576 feet.

Question 10: What is the total of the series' first 50 terms: 2 + 17 + 32 + 47 +?

18,475 (Correct answer)
19,125
18,235
1,600

Correct answer: 18,475

This is an arithmetic series with a first term (a_1) of 2 and a common difference (d) of 17 - 2 = 15. To find the sum of the first 50 terms (S_50), use the formula S_n = n/2 * (2a_1 + (n-1)d). Plugging in the values, S_50 = 50/2 * (2*2 + (50-1)*15) = 25 * (4 + 49*15) = 25 * (4 + 735) = 25 * 739 = 18,475.

Question 11: -3, -9, -27, -81, -27, find the common ratio.

r = -3
r = 6
r = -6
r = 3 (Correct answer)

Correct answer: r = 3

The common ratio (r) in a geometric sequence is found by dividing any term by its preceding term. For this sequence, -9 / -3 = 3. Checking the next terms, -27 / -9 = 3 and -81 / -27 = 3, confirms the common ratio is 3.

Question 12: Identify the common ratio between 16, 8, 4, and 2.

r = 8
r = 1/4
r = 1/2 (Correct answer)
r = 2

Correct answer: r = 1/2

The common ratio (r) in a geometric sequence is found by dividing any term by its preceding term. For this sequence, 8 / 16 = 1/2. Checking the next terms, 4 / 8 = 1/2 and 2 / 4 = 1/2, confirms the common ratio is 1/2.

Question 13: Determine the infinite total of the sequence Aₙ = 2(1/3)ⁿ

S = 3
S = 2
S = 1 (Correct answer)
S is not finite

Correct answer: S = 1

The given sequence is Aₙ = 2(1/3)ⁿ. The first term (a) is A₁ = 2(1/3)¹ = 2/3. The common ratio (r) is 1/3, as it's the base of the exponent n. Since |r| = 1/3 < 1, the infinite sum exists and can be calculated using S = a / (1 - r) = (2/3) / (1 - 1/3) = (2/3) / (2/3) = 1.

Question 14: What is the result of Sₙ = (n/2)(A1 + Aₙ)?

The nth partial sum of any series.
The nth partial sum of a geometric series.
The infinite sum of absolutely any series in the whole wide world.
The nth partial sum of an arithmetic series. (Correct answer)

Correct answer: The nth partial sum of an arithmetic series.

The formula Sₙ = (n/2)(A₁ + Aₙ) is a fundamental equation used to calculate the sum of the first 'n' terms of an arithmetic series. It works by averaging the first and last terms and multiplying by the number of terms. This formula is specific to arithmetic sequences, not geometric or general series.

Question 15: Determine the first six terms in the sequence: aₙ = (n + 1)³

8, 27, 64, 125, 216, 343 (Correct answer)
6, 9, 14, 21, 30, 41
3, 6, 9, 12, 15, 18
None of the above

Correct answer: 8, 27, 64, 125, 216, 343

To find the first six terms, substitute n = 1, 2, 3, 4, 5, and 6 into the formula aₙ = (n + 1)³. This yields: (1+1)³=8, (2+1)³=27, (3+1)³=64, (4+1)³=125, (5+1)³=216, and (6+1)³=343. These are the first six terms of the sequence.

Question 16: What does the equation S = A₁/(1-r) yield?

The partial sum of a geometric series.
The infinite sum of an arithmetic series.
The infinite sum of a geometric series. (Correct answer)
The finite sum of an arithmetic series.

Correct answer: The infinite sum of a geometric series.

The equation S = A₁/(1-r) is the standard formula for calculating the sum of an infinite geometric series. This formula is applicable only when the absolute value of the common ratio (r) is less than 1 (|r| < 1), ensuring that the series converges to a finite sum.

Free Pre-Calculus Sequences and Series Questions and Answers

Create the geometric sequence's formula: -2, 10, -50, 250...