How do you solve this? The host's doorbell rang 20 times. The first time the doorbell rang, only one guest arrived. Each time the doorbell rang after that, two more guests arrived than had arrived on the previous ring. How many guests arrived in all at the party?
This is an arithmetic sequence, where your terms are
1, 3, 5, 7, ...
a = 1, d = 2
you want
sum(20)
= ..... , sub into your formula for sum(n) of an AS
To solve this problem, we can use algebraic thinking to set up an equation.
Let's assume that on the first ring, only one guest arrived.
On the second ring, two more guests arrived than on the previous ring. This means that on the second ring, 1 + 2 = 3 guests arrived.
On the third ring, two more guests arrived than on the previous ring. So on the third ring, 3 + 2 = 5 guests arrived.
We can see a pattern here: the number of guests increases by 2 each time the doorbell rings.
Let's represent the number of guests arriving on each ring with variables. We'll use "x" to represent the number of guests on the first ring.
On the first ring, x guests arrived.
On the second ring, x + 2 guests arrived.
On the third ring, (x + 2) + 2 guests arrived.
And so on...
Now, we are given that the doorbell rang 20 times. We want to find the total number of guests who arrived at the party.
To find the total number of guests, we need to sum up the number of guests on each ring. We can do this by setting up an equation.
The sum of the number of guests on each ring can be expressed as:
x + (x + 2) + ((x + 2) + 2) + ... + ((x + 2) + 2(n-1))
Here, n represents the number of rings. Since there were 20 rings, n = 20.
Now we can substitute the value of n into the equation:
x + (x + 2) + ((x + 2) + 2) + ... + ((x + 2) + 2(n-1))
= x + (x + 2) + ((x + 2) + 2) + ... + ((x + 2) + 2(20-1))
= x + (x + 2) + ((x + 2) + 2) + ... + ((x + 2) + 38)
Now we can simplify the equation:
2n + x + x + 2n-2 + x + 2n-4 + ... + x + 2
= 2(20) + x + x + 2(19) + x + 2(18) + ... + x + 2
= 40 + x + x + 38 + x + 36 + ... + x + 2
Since there are 20 terms in the equation, we can rewrite it as:
20x + 2(1 + 2 + 3 + ... + 19)
To find the sum of the numbers from 1 to 19, we can use the formula for the sum of an arithmetic series:
S = (n/2)(first term + last term)
In this case, n = 19 and the first term is 1, and the last term is 19.
S = (19/2)(1 + 19)
= (19/2)(20)
= 190
Now we can substitute this value back into the original equation:
20x + 2(190)
= 20x + 380
We know that the total number of rings is given as 20, so we can set up another equation:
20x + 380 = 20
Solving this equation will give us the value of x, which represents the number of guests on the first ring.
20x = 20 - 380
20x = -360
x = -360/20
x = -18
Since we cannot have a negative number of guests, this solution does not make sense in the context of the problem. Thus, there seems to be an error in the problem statement or logic.
Please double-check the problem statement or provide any additional information if available.