find all sets of three consecutive odd integers whose sum is greater than 54 and less than 63
The sum of three consecutive odd integers is
S(n)=(n-2) + n + (n+2) = 3n
So 54<S(n)<63, or
54 < 3n < 63
Divide all by 3 to get
18 < n < 21
Solve for all valid (integer) value of n.
Each value of n relates to the set of integers {n-2, n, n+2} that satisfies the specified requirements.
how do you get n-2, n, n+2 I keep getting n, n+2 and n+4?
To find all sets of three consecutive odd integers whose sum is greater than 54 and less than 63, we can try different combinations and check if they meet the given criteria.
Let's assume the first odd integer is represented by n.
The next two odd integers would then be (n + 2) and (n + 4) since they are consecutive.
To find the sets that meet the criteria:
1. Start with n = 1 (smallest possible odd integer)
2. Substitute the values into the equation: n + (n + 2) + (n + 4) > 54 and n + (n + 2) + (n + 4) < 63
3. Simplify the equation:
3n + 6 > 54 and 3n + 6 < 63
3n > 48 and 3n < 57
n > 16 and n < 19
Since n has to be an odd integer, n can be either 17 or 18.
For n = 17:
The set of three consecutive odd integers is 17, 19, and 21.
The sum of these integers is 17 + 19 + 21 = 57, which is within the specified range.
For n = 18:
The set of three consecutive odd integers is 19, 21, and 23.
The sum of these integers is 19 + 21 + 23 = 63, which exceeds the upper limit of 63.
Therefore, the only set of three consecutive odd integers that meets the criteria is 17, 19, and 21.
To find all sets of three consecutive odd integers whose sum is greater than 54 and less than 63, we can set up an equation and solve for the values.
Let's denote the first odd integer as x. Since we are looking for consecutive odd integers, we can express the other two integers as x + 2 and x + 4.
Now we can create an equation using the sum of the three consecutive odd integers:
x + (x + 2) + (x + 4) > 54
3x + 6 > 54
3x > 48
x > 16
Now, let's find the upper limit.
x + (x + 2) + (x + 4) < 63
3x + 6 < 63
3x < 57
x < 19
Therefore, the first odd integer must be greater than 16 and less than 19.
Let's list out the values to find all the sets that meet the criteria:
When x = 17, the three consecutive odd integers are: 17, 19, 21.
When x = 18, the three consecutive odd integers are: 19, 21, 23.
So, the sets of three consecutive odd integers whose sum is greater than 54 and less than 63 are {17, 19, 21} and {19, 21, 23}.