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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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."
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.
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.
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.
