This question has been discussed many times.

To point a few,

https://www.beatthegmat.com/a-new-approa ... 17860.html
https://www.beatthegmat.com/if-each-term ... 13856.html

This question has been answered by atleast three experts in two diff threads. Don't know where it is appropiate one to air my doubts so I have continued in the recent thread. Most of them have analysed the problem and its nuances rather than solving it algeberically. I find that it difficult to conceive the problem so minutely when the time available is under 2 mins. Is it better to solve it rather brutishly by simultaneous eq. in the above problem

the number are either 7 or 77 so there are certain nos of 7 and 77 say x and y respectively
7x+77y = 350 or
x + 11 y = 50 ( eq 1) this is similar to the approach in post of lunarpower.
x + y = n (eq 2) n is the no of terms in the series. subtracting 2 from 1
10y = 50-n of the options A. 38 B. 39 C. 40 D. 41 E. 42
only n = 40 will result in y being an integer.

Let x and y be number of 7s and 77s respectively

7x + 77y = 350
x +11y = 50
also
x +y = n
substitute one equation in another to obtain

10(5-y) = n

That implies n has to be a multiple of 10

Hence choice C