The sum of the given sequence in sigma notation is:
∑(n=1 to 18519) 6n-3 and the sum of the sequence is 203704664.
The given sequence has a common difference of 6. Therefore, we can find the nth term using the formula:
nth term = a + (n-1)d
where a is the first term and d is the common difference.
Here, a = 3 and d = 6. Thus, the nth term is:
nth term = 3 + (n-1)6 = 6n-3
To find the sum of the sequence, we can use the formula for the sum of an arithmetic series:
Sum = n/2(2a + (n-1)d)
where n is the number of terms.
Here, a = 3, d = 6, and the last term is 111111. We need to find n, the number of terms:
111111 = 6n-3
6n = 111114
n = 18519
Therefore, there are 18519 terms in the sequence.
Substituting the values in the formula, we get:
Sum = 18519/2(2(3) + (18519-1)6) = 203704664
Thus, the sum of the given sequence in sigma notation is:
∑(n=1 to 18519) 6n-3 and the sum of the sequence is 203704664.
Know more about sum of an arithmetic series here:
https://brainly.com/question/14203928
#SPJ11
An aeroplane has 30 seats. 95% of people show up for their journey. You have been hired by the travel company to recommend how many tickets they sell for the aeroplane. Stating your assumptions clearly and explaining the risk to the company of having a passenger who can't get on the plane show how many tickets you would sell.
It is important for the airline to sell the correct number of tickets to avoid such scenarios.
An airplane with 30 seats has to sell its tickets in such a way that it doesn't go empty and doesn't carry any overcapacity. To calculate how many tickets should be sold, we need to make some assumptions. For this purpose, the following assumptions are made:AssumptionsAssuming that the number of passengers is large and statistically significant, it is safe to assume that 95% of passengers will show up for their journey. The airline has no way to predict which specific passenger will miss their flight and is dependent on historical data.
The airline will provide a 100% refund for passengers who miss their flights. The airline will make no profit on these tickets sold and will only cover their costs.The probability that at least one passenger will not show up for their journey is 5%.There is a chance that all passengers might show up for their flight. If this happens, the airline may face a penalty for overselling the airplane seats.The number of tickets the airline sells is the sum of the expected number of passengers and some additional seats as a safety buffer to account for the cases where all passengers show up for their journey.
The probability that all passengers show up for their flight is calculated as follows:P(all passengers show up) = P(First passenger shows up) * P(Second passenger shows up) * … * P(Last passenger shows up) = 0.95^30 = 0.00276 = 0.276%This means there is only a 0.276% chance that all passengers will show up for their journey. Therefore, the airline should sell the expected number of passengers plus a safety buffer to account for this scenario. Expected passengers = 30 * 0.95 = 28.5 passengers Therefore, the number of tickets the airline should sell is 29. The extra seat serves as a buffer, protecting the airline from financial penalties if all passengers show up.
What is the risk to the company of having a passenger who can't get on the plane?If the company sells 30 tickets and all passengers show up, then one passenger will not be able to board the plane. This may cause a delay in the flight and impact customer satisfaction. In addition, the airline may face a penalty for overselling seats. This can lead to financial losses for the airline. Therefore, it is important for the airline to sell the correct number of tickets to avoid such scenarios.
Learn more about Probability here,https://brainly.com/question/13604758
#SPJ11
Write an R program that simulates a system of n components
connected in parallel. Let the probability that a component fails
be p (use p = 0.01). Estimate the probability that the system
fails.
The program that simulates a system of n components connected in parallel is coded below.
The R program that simulates a system of n components connected in parallel and estimates the probability that the system fails, given the probability that a component fails (p):
simulate_parallel_system <- function(n, p) {
num_trials <- 10000 # Number of trials for simulation
num_failures <- 0 # Counter for system failures
for (i in 1:num_trials) {
system_fail <- FALSE
# Simulate each component
for (j in 1:n) {
component_fail <- runif(1) <= p # Generate a random number and compare with p
if (component_fail) {
system_fail <- TRUE # If any component fails, system fails
break
}
}
if (system_fail) {
num_failures <- num_failures + 1
}
}
probability_failure <- num_failures / num_trials
return(probability_failure)
}
# Usage example
n <- 10
p <- 0.01
probability_system_failure <- simulate_parallel_system(n, p)
print(paste("Estimated probability of system failure:", probability_system_failure))
In this program, the `simulate_parallel_system` function takes two parameters: `n` (the number of components in the system) and `p` (the probability that a component fails). It performs a simulation by running a specified number of trials (here, 10,000) and counts the number of system failures. The probability of system failure is estimated by dividing the number of failures by the total number of trials.
Learn more about probability here:
https://brainly.com/question/17463486
#SPJ4
Let F(x,y,z)=yzi+xzj+(xy+1)k be a vector field. (i) Find a potential ϕ(x,y,z) such that F=∇ϕ and ϕ(0,0,0)=2. Ans: xyz+z+2 (ii) Let C be a curve with parametrization r(t),0≤t≤2. Suppose, r(0)=(0,0,0),r(1)= (1,1,1) and r(2)=(2,2,2). Find ∫CF⋅dr.
The potential ϕ(x,y,z) for the vector field F(x,y,z)=yzi+xzj+(xy+1)k is ϕ(x,y,z) = xyz+z+2.
To find the line integral ∫CF⋅dr, we need to evaluate the dot product of F and dr along the curve C. Given that r(t) is the parametrization of C, we can express dr as dr = r'(t)dt.
Substituting the values of r(t) into F(x,y,z), we get F(r(t)) = (tz, t, t^2+1). Taking the dot product with dr = r'(t)dt, we have F(r(t))⋅dr = (tz, t, t^2+1)⋅(dx/dt, dy/dt, dz/dt)dt.
Now we substitute the values of r(t) and r'(t) into the dot product expression and integrate it over the given range of t, which is 0≤t≤2. This will give us the value of the line integral ∫CF⋅dr.
Since the specific values of dx/dt, dy/dt, and dz/dt are not provided, we cannot calculate the exact value of the line integral without additional information.
To learn more about integral click here
brainly.com/question/31433890
#SPJ11
A bag contains 19 red balls, 7 blue balls and 8 green balls. a) One ball is chosen from the bag at random. What is the probability that the chosen ball will be blue or red? Enter your answer as a fraction. b) One ball is chosen from the bag at random. Given that the chosen ball is not red, what is the probability that the chosen breen? Enter your answer as a fraction.
A) The probability that the chosen ball will be blue or red is 2/3.b) The probability that the chosen ball will be green given that it is not red is 8/15.
a) One ball is chosen at random from the bag. The probability that the ball chosen will be blue or red can be calculated as follows:
We have 19 red balls and 7 blue balls. So, the total number of favourable outcomes is the sum of the number of red balls and blue balls.i.e, the total number of favourable outcomes = 19 + 7 = 26
Also, there are 19 red balls, 7 blue balls and 8 green balls in the bag.
So, the total number of possible outcomes = 19 + 7 + 8 = 34
Therefore, the probability that the ball chosen will be blue or red is given by:
Probability of blue or red ball = (Number of favourable outcomes) / (Total number of possible outcomes)
Probability of blue or red ball = (26) / (34)
Simplifying the above fraction gives us the probability that the chosen ball will be blue or red as a fraction i.e.2/3
b) One ball is chosen at random from the bag. Given that the chosen ball is not red, we have only 7 blue balls and 8 green balls left in the bag.So, the total number of favourable outcomes is the number of green balls left in the bag, which is 8.
Therefore, the probability that the chosen ball is green given that it is not red is given by:
Probability of green ball = (Number of favourable outcomes) / (Total number of possible outcomes)
Probability of green ball = 8 / 15
Simplifying the above fraction gives us the probability that the chosen ball will be green as a fraction i.e.8/15.
The final answers for the question are:a) The probability that the chosen ball will be blue or red is 2/3.b) The probability that the chosen ball will be green given that it is not red is 8/15.
Know more about probability here,
https://brainly.com/question/31828911
#SPJ11