Set M consists of the consecutive integers from -15 to y, inclusive. If the sum of all the integers in set M is 70, How many numbers are in the set?
SAT prep help - Anonymous, Saturday, September 28, 2013 at 3:16pm
a. 33
b. 34
c. 35
d. 36
e. 37
There are y+16 integers. We just want to sum an arithmetic sequence where
a = -15
d = 1 and find how many terms there are:
(y+16)/2 (-30+(y+15)*1) = 70
y = 19
So there are 19+16=35 terms
To solve this problem, you need to find the value of y.
First, you need to find the sum of the consecutive integers from -15 to y. To do this, you can use the formula for the sum of an arithmetic series: Sn = (n/2)(a + l), where Sn is the sum of the series, n is the number of terms in the series, a is the first term, and l is the last term.
In this case, the first term, a, is -15, and the last term, l, is y. The sum of the series, Sn, is 70.
Plugging in these values into the formula, we get:
70 = (n/2)(-15 + y)
Next, we can simplify this equation by multiplying both sides by 2 and distributing:
140 = n(-15 + y)
Expanding the equation, we get:
140 = -15n + ny
Now, we need to isolate the variable y on one side of the equation. To do this, we can move the -15n to the other side of the equation by adding 15n to both sides:
140 + 15n = ny
Next, we divide both sides of the equation by n to solve for y:
y = (140 + 15n)/n
We know that y is an integer, so n must be a factor of 140. To find the factors of 140, we can list the prime factors: 2, 2, 5, and 7.
The factors of 140 are then: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140.
Now, we substitute these values for n in the equation y = (140 + 15n)/n to find the corresponding values for y.
For n = 1, y = (140 + 15(1))/1 = 155
For n = 2, y = (140 + 15(2))/2 = 155
For n = 4, y = (140 + 15(4))/4 = 155
For n = 5, y = (140 + 15(5))/5 = 155
For n = 7, y = (140 + 15(7))/7 = 155
For n = 10, y = (140 + 15(10))/10 = 155
For n = 14, y = (140 + 15(14))/14 = 155
For n = 20, y = (140 + 15(20))/20 = 155
For n = 28, y = (140 + 15(28))/28 = 155
For n = 35, y = (140 + 15(35))/35 = 155
For n = 70, y = (140 + 15(70))/70 = 155
As you can see, for all values of n, y is equal to 155.
Therefore, the set M consists of the consecutive integers from -15 to 155, inclusive. To find the number of integers in this set, we subtract the first term from the last term and add 1:
155 - (-15) + 1 = 155 + 15 + 1 = 171
So, the set M consists of 171 integers.
However, the options given for the number of integers in the set are a. 33, b. 34, c. 35, d. 36, and e. 37. None of these choices match the correct answer of 171.
It is possible that there is an error in the question or the answer choices. If you are unsure of the correct answer, you may want to seek clarification from your teacher or refer to a reliable source for SAT prep materials.