Which of the following represents the general term for the sequence 2, 4, 6, 8, 10, . . .?

To work with the repeating decimals, note that since there are two digits repeating, you will be using a 1 followed by two zeros, which is 100. This factor will be important for subsequent calculations.

Let’s denote the repeating decimal as x.

Thus, you can write

x = 0.212121...

Now, if you multiply this equation by 100, you can write it on top of the previous equation.
This results in:

100x = 21.212121...
x = 0.212121...

Next, set up a subtraction by taking the second equation from the first:

100x = 21.212121...
x = 0.212121...
- ----------------------------
99x = 21

Solving for x involves dividing both sides by 3:

33x = 7

Then, divide both sides by 33:

x = 7/33

Thus, we conclude that the repeating decimal 0.212121... equates to the fraction 7/33.

Answer: the fraction is 7/33

To determine the future value using simple interest, the formula is: future value = present value (1+n*rate), where n represents the number of years and rate indicates the annual interest as a decimal. By substituting the values into the equation, we find that 10000 = 3900(1 + 0.0783*t). Therefore, we calculate 1 + 0.0783t = 10000/3900, leading to t = (10000/3900 - 1)/0.0783, which equals approximately 19.98 years. For compound interest, the future value is determined by present value (1 + rate/n)^(nt), where n=1, rate=0.0783, future value=10000, present value=3900, ultimately resulting in t = log(10000/3900)/log(1.0783) which yields roughly 12.49 years.

Given: Triangle ABC is rotated 90 degrees counterclockwise and then translated vertically by 3 units to form triangle A'B'C'. To determine: The transformation rule necessary to map each point (x,y) on Triangle ABC to its equivalent point on Triangle A'B'C'. Solution: For a figure that undergoes a 90-degree counterclockwise rotation, the transformation will be defined as: To translate a figure upward by 3 units, the appropriate transformation will be applied afterwards. Hence, when triangle ABC experiences a rotation of 90 degrees counterclockwise followed by a translation of 3 units upward to create triangle A'B'C', the transformation rule will be established accordingly.