Mina Satti wrote:
Find m if the sum of the consecutive integers from -16 to m, inclusive is 74.
A) 17
B) 18
C) 19
D) 20
E) 21
We can use the following formula:
sum = average x quantity
Since we have a set of evenly spaced integers, the average is (-16 + m)/2.
We can also calculate the quantity as m - (-16) + 1 = m + 17.
Thus:
74 = (-16 + m)/2 x (m + 17)
148 = (m - 16)(m + 17)
148 = m^2 + m - 272
m^2 + m - 420 = 0
(m + 21)(m - 20) = 0
m = -21 or m = 20
Since m > -16, m must be 20.
Alternate Solution:
The integers in the sum will be -16, -15, -14, …, -2, -1, 0, +1, +2, …, m. Notice that all the negative integers from -16 to -1 inclusive will be offset by the positive integers from +1 to +16, yielding a net sum of 0. So now, starting with the integer +17, we need to add consecutive integers until we get a sum of 74. We see that 17 + 18 + 19 + 20 = 74, and thus m = 20, the largest integer in the set.
Answer: D
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