1. The answer is C. 124. The midpoint of one side of our square is 4. So we take the two consecutive sides of the square to form another side of the next square. However, to get the measure of the side of the new square, we solve for it using the Pythagorean Theorem which states that the square of the hypotenuse is equal to the square of the sides of a right triangle c^2= a^2+ b^2.
The measure of the new side of our square is now square root of 32 or 4 square of 2. We take the midpoint of the new side. We divide by 2, and the answer is 2 square root of 2. From there we use it as a new side of another right triangle so that we can solve again the measure of the hypotenuse which will be the new side of our square. We repeat the process until we get the 6th square.
2. The answer is 31. If we make an illustration, we get 31 text messages by tree diagram. However, we can also use the concept of a geometric sequence, and geometric series. We need to find our first term, which is 1, and our common ratio, r, which is two. Also, we need the n which is the number of terms involved, which is 5. Using the formula of a geometric series S_n= (a_1 (r^n-1))/(r-1), our answer is 31.
