The least integer of a set of consecutive integers is -48. if the sum of these integers is 49, how many integers are in the set?
The sum of consecutive integers from -48 to +48 is zero.
If the sum is 49, the sequence finishes at +49.
How many integers are there from -48 to +49?
98
To find the number of integers in the set, we need to determine the range of the consecutive integers.
The least integer in the set is -48, and let's assume the number of integers in the set is n.
The sum of the set of consecutive integers can be found using the formula: sum = (first term + last term) * number of terms / 2
In this case, the first term is -48, the last term can be calculated by starting from -48 and increasing by 1 each time until we reach the last term, so the last term would be (-48 + n - 1), and the sum is given as 49.
Therefore, we have the equation: 49 = (-48 + (-48 + n - 1)) * n / 2
Simplifying the equation: 49 = (-96 + n) * n / 2
Multiplying both sides by 2 to eliminate the fraction: 98 = (-96 + n) * n
Expanding and rearranging the equation: n^2 - 96n + 98 = 0
We can solve this quadratic equation using the quadratic formula: n = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -96, and c = 98. Plugging these values into the quadratic formula:
n = (-(-96) ± √((-96)^2 - 4(1)(98))) / (2 * 1)
n = (96 ± √(9216 - 392)) / 2
n = (96 ± √8824) / 2
The discriminant (√8824) is approximately equal to 93.89, so we have two possible values for n:
n ≈ (96 + 93.89) / 2 ≈ 189.89 / 2 ≈ 94.94 (approximately)
n ≈ (96 - 93.89) / 2 ≈ 2.11 / 2 ≈ 1.06 (approximately)
Since the number of integers cannot be a fraction or a decimal, we round down the value of n to the nearest whole number.
Therefore, there are approximately 94 integers in the set.
To find the number of integers in the set, we need to determine the range of the set and divide it by the common difference of the consecutive integers.
Let's denote the common difference as 'd', and the number of integers in the set as 'n'.
The least integer in the set is -48, so the next consecutive integer would be -48 + d.
Since the sum of the integers is 49, we can set up the following equation:
-48 + (-48 + d) + (-48 + 2d) + ... + (-48 + (n-1)d) = 49
Now, let's simplify this equation:
-n * 48 + d (1 + 2 + ... + n-1) = 49
We know that the sum of the first n natural numbers can be calculated using the formula: 1 + 2 + ... + n = (n * (n + 1)) / 2. Substituting this into the equation, we have:
-n * 48 + d * ((n * (n + 1)) / 2) = 49
Further simplifying:
-n * 48 + (n * (n + 1)) * d / 2 = 49
Now, we can use the equation to find the value of 'n'. One approach is to check the possible values for 'n' and see which one satisfies the given equation.
Let's start with 'n' = 1:
-1 * 48 + (1 * (1 + 1)) * d / 2 = 49
-48 + 2d/2 = 49
-48 + d = 49
d = 97
However, 'd' cannot be 97 because the integers are consecutive, and the difference between consecutive integers must be constant. So, 'n' = 1 does not satisfy the equation.
Let's try 'n' = 2:
-2 * 48 + (2 * (2 + 1)) * d / 2 = 49
-96 + 3d/2 = 49
3d/2 = 145
3d = 290
d = 290/3
Since 'd' is not an integer, 'n' = 2 does not satisfy the equation.
Let's try 'n' = 3:
-3 * 48 + (3 * (3 + 1)) * d / 2 = 49
-144 + 6d/2 = 49
-144 + 3d = 49
3d = 193
d = 193/3
Again, 'd' is not an integer, so 'n' = 3 does not satisfy the equation.
By continuing this process, we find that for 'n' = 5, we get an integer value for 'd':
-5 * 48 + (5 * (5 + 1)) * d / 2 = 49
-240 + 15d/2 = 49
-240 + 15d = 98
15d = 338
d = 338/15 = 22.53...
Since 'd' is not an integer, 'n' = 5 does not satisfy the equation.
Therefore, it seems that there is no integer value for 'n' that satisfies the equation.