What is the 1st term in sequence S?

What is the 1st term in sequence S?

(1) The 3rd term of S is 4.
(2) The 2nd term of S is three times the 1st, and the 3rd term is four times the 2nd.

(1) is no help in finding the first term of S. For example, the following sequences each have 4 as their third term, yet they have different first terms:

0, 2, 4
-4, 0, 4

This eliminates choices A and D. Now, even if we are unable to solve this problem, we have significantly increased our chances of guessing correctly--from 1 in 5 to 1 in 3.

Turning to (2), we completely ignore the information in (1). Although (2) contains a lot of information, it also is not sufficient. For example, the following sequences each satisfy (2), yet they have different first terms:

1, 3, 12
3, 9, 36

This eliminates B, and our chances of guessing correctly have increased to 1 in 2.

Next, we consider (1) and (2) together. From (1), we know "the 3rd term of S is 4." From (2), we know "the 3rd term is four times the 2nd." This is equivalent to saying the 2nd term is 1/4 the 3rd term: (1/4)4 = 1. Further, from (2), we know "the 2nd term is three times the 1st." This is equivalent to saying the 1st term is 1/3 the 2nd term: (1/3)1 = 1/3. Hence, the first term of the sequence is fully determined: 1/3, 1, 4. The answer is C.

In a triangle ABC, AB = 6, BC = 8.  What is the area of the square.

(1) AC = AB square + BC square.
(2) Angle ABC = 90 deg

Recall that a triangle is a right triangle if and only if the square of the longest side is equal to the sum of the squares of the shorter sides (Pythagorean Theorem). Hence, (1) implies that the triangle is a right triangle. So the area of the triangle is (6)(8)/2. Note, there is no need to calculate the area--we just need to know that the area can be calculated. Hence, the answer is either A or D.

Turning to (2), we see immediately that we have a right triangle. Hence, again the area can be calculated. The answer is D.