Problem 1:

2^n = n^2.

n has to be a multiple of 2 (more specifically a power of 2).

n = 2 and n = 4 satisfy the conditions. For all other values of n = 2^x, we
can see that the difference between the terms keeps increasing. So,
only 2 values will satisfy the condition.

Alternatively, take n = 2^t.
2^n = (2^t)^2 = 2^2t
So, n = 2t = 2^t. Only t=1 and t=2 will satisfy this.

----------------------------------------------------------------------------------------

Problem 2:

We know there can be a max of 4 77's in the sequence - else the sum
will be greater than 350. Try 77*2.

77*2 + 7*28 = 350 and we have 30 terms. So, this can't be it.

We can see that as we increase the number of 77s, the total number of
terms i.e. n will go down. So, try 77*1

77*1 + 7*39 = 350. Total of 40 terms...