350/7 = 50.If each term in the sum a1+a1+a3+a4+...+aN is either 7 or 77 and the sum equals 350, which of the following could be equal to n?
1) 38
2) 39
3) 40
4) 41
5) 42
So if each term were 7, there would be 50 terms.
The answer choices are all a little less than 50.
Implication:
Most -- but not all -- of the terms will be 7.
Case 1: 1 term = 77
Amount remaining = 350-77 = 273.
273/7 = 39
This works!
Thus, 1 term = 77, while 39 terms = 7, for a total of 40 terms.
The correct answer is C.
Alternate approach:
Let x = the number of 7's and y = the number of 77's.
Since the sum is 350, we get:
7x + 77y = 350
x + 11y = 50.
The answer choices imply that the total number of terms -- x+y -- is between 38 and 42, inclusive.
Thus:
38 ≤ x+y ≤ 42.
Substituting x = 50-11y into 38 ≤ x+y ≤ 42, we get:
38 ≤ (50-11y) +y ≤ 42
38 ≤ 50 - 10y ≤ 42
-12 ≤ - 10y ≤ -8
12 ≥ 10y ≥ 8.
Only one integer value satisfies the resulting inequality:
y = 1.
Since y=1 and x=50-11y, we get:
x = 50 - 11*1 = 39.
Thus, the total number of terms = x+y = 39+1 = 40.
