FREE Pre-Calculus Sequences and Series Questions and Answers
Determine the sum, if it exists, of the infinite geometric series: 200 - 100 + 50 - 25 +
The sum of an infinite geometric series can be calculated using the formula S = a / (1 - r), where "a" is the first term and "r" is the common ratio. In the given series 200 - 100 + 50 - 25 + ..., the first term "a" is 200, and the common ratio "r" is -1/2. Plugging these values into the formula, we get S = 200 / (1 - (-1/2)) = 200 / (3/2) = 400/3. Therefore, the correct answer is 400/3.
Determine the common difference 97, 86, 75, 64, ...
The common difference in an arithmetic sequence represents the constant value added to each term to get to the next term. In this case, since the sequence is decreasing, the common difference is negative. By observing the terms, we can see that each term is decreasing by 11 units to get to the next term. Therefore, the correct answer is -11.
What does the equation S = A₁/(1-r) yield?
The equation S = A₁/(1-r) yields the infinite sum of a geometric series. In this formula, S represents the sum to infinity, A₁ is the first term of the geometric series, and r is the common ratio between consecutive terms. By using this equation, one can calculate the total sum of all the terms in the geometric series as the number of terms approaches infinity.
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?
In six seconds, the object will fall a total of 576 feet. This can be calculated by adding up the distances fallen in each second: 16 + 48 + 80 + 112 + 144 + 176 = 576 feet.
Determine the first six terms in the sequence: aₙ = (n + 1)³
The first six terms in the sequence aₙ = (n + 1)³ are 8, 27, 64, 125, 216, and 343. These values are obtained by substituting n = 1, 2, 3, 4, 5, and 6 into the formula (n + 1)³, resulting in the respective terms of the sequence.
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?
To find the sum of the first ten terms of a geometric sequence, we first need to find the common ratio "r." We can do this by dividing the fifth term (1536) by the third term (96), which gives us a common ratio of 16. Then, we use the formula for the sum of the first "n" terms of a geometric sequence, S = a * (1 - r^n) / (1 - r), where "a" is the first term and "n" is the number of terms. Plugging in the values, we get S = 96 * (1 - 16^10) / (1 - 16) = 2,097,150. Therefore, the correct answer is 2,097,150.
What is the total of the series' first 50 terms: 2 + 17 + 32 + 47 +?
The total of the first 50 terms in the series can be calculated by finding the arithmetic progression. The common difference between consecutive terms is 15 (17 - 2 = 15, 32 - 17 = 15, and so on). Using the formula for the sum of an arithmetic progression (S = (n/2) * (a + l)), where n is the number of terms (50), a is the first term (2), and l is the last term (l = a + (n-1) * d = 2 + 49 * 15 = 747), we get S = (50/2) * (2 + 747) = 25 * 749 = 18,475. Therefore, the total of the series' first 50 terms is 18,475.
Determine the infinite total of the sequence Aₙ = 2(1/3)ⁿ
The infinite total (sum to infinity) of the sequence Aₙ = 2(1/3)ⁿ is 1. As the value of n approaches infinity, the terms in the sequence get closer and closer to zero, and the sum of all these infinitely decreasing terms converges to 1.
-3, -9, -27, -81, -27, find the common ratio.
The common ratio in this geometric sequence is 3. This can be determined by observing that each term is obtained by multiplying the previous term by -3, indicating a common ratio of -3. However, the last term seems to be mistakenly written as -27 instead of -243 (as per the pattern), but despite this inconsistency, the common ratio remains 3.
Determine the sum of the terms in the sequence 1, 1/2, 1/4, 1/8, and 1/16...
The sum of the terms in a geometric sequence can be calculated using the formula S = a / (1 - r), where "a" is the first term and "r" is the common ratio. In the given sequence 1, 1/2, 1/4, 1/8, 1/16, the first term "a" is 1, and the common ratio "r" is 1/2. Plugging these values into the formula, we get S = 1 / (1 - 1/2) = 1 / (1/2) = 2. Therefore, the correct answer is 2.
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?
To find out in how many weeks you will be able to jog for 60 minutes per day, we can use arithmetic progression. The initial time is 12 minutes, and each subsequent week increases by 6 minutes. We need to find when the jogging time reaches 60 minutes. Solving for n in the equation 12 + 6(n - 1) = 60, we get n = 9. Therefore, it will take 9 weeks for you to be able to jog for 60 minutes per day.
What is the result of Sₙ = (n/2)(A1 + Aₙ)?
The result of Sₙ = (n/2)(A1 + Aₙ) is the nth partial sum of an arithmetic series. In this formula, A1 represents the first term, Aₙ represents the nth term, and n is the number of terms in the series. It calculates the sum of the first n terms in an arithmetic series.
Identify the common ratio between 16, 8, 4, and 2.
The common ratio in this geometric sequence is 1/2. This can be determined by observing that each term is obtained by multiplying the previous term by 1/2, indicating a common ratio of 1/2.
Create the geometric sequence's formula: -2, 10, -50, 250...
The formula for the geometric sequence is given as aₙ = -2(-5)^(n-1), where "aₙ" represents the nth term of the sequence. This formula allows us to find any term in the sequence by plugging in the appropriate value of "n." Therefore, the correct answer is aₙ = -2(-5)^n-1.
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?
The sequence representing the number of carrot cakes produced by Eliya's Bakery is an arithmetic sequence. In an arithmetic sequence, each term is obtained by adding a constant value (common difference) to the previous term. Here, the number of carrot cakes is decreasing by a constant value of 17 each month, resulting in an arithmetic sequence. Therefore, the correct answer is "Arithmetic."
Determine the 22nd term of the sequence 5, 8, 11,...
To find the 22nd term of the sequence 5, 8, 11,..., we can observe that each term is obtained by adding 3 to the previous term. Starting with the first term (5) and adding 3 repeatedly, we can calculate the 22nd term as 5 + 3*(22-1) = 5 + 3*21 = 68. Therefore, the correct answer is 68.