to find the integers you can guess and check (you know both are larger than 10 because their product is greater than 100) or you can set up a system of equations.  if a is the larger number and .

Therefore:

if you solve that quadratic you get 

 and b is the smaller number so the bigger number is 12

For a problem like this, you can always use the answers to find your correct answer. By choosing each number, you can find the other two options and then add together your values. You would, for instance, take  and say, "The list must be: ." Then, adding them to get , you will know that this is not correct.

However, you can do this much more easily with algebra. You know that five consecutive integers are going to look like:

, where  is the height of the shortest person. Thus, you know that the total inches of the students can be represented in the following manner:

This simplifies to:

Solving for , you get:

However, remember that you need to find the second tallest person. This means that your list is: . Thus, your answer is .

An odd integer can be expressed as  because two times any number is an even number and one plus an even number is always odd. We can then write these three consecutive odd integers in terms of  as  . We can then square each of these numbers and add them together. 

Then use binomial expansion to rewrite the expression (better known as FOIL).

We can then combine like terms and set it equal to  as given.

This tells us that two possible sets of numbers satisfy this condition:  and . It is evident that the sums of the squares of these numbers should be the same, so we cannot determine which set the question is discussing.

A geometric sequence is one where two get two each consecutive number in the sequence, you must multiply or divide a number.  If we look at the sequence, we can see that the pattern is dividing by  each time.  Therefore, to get the next term in the sequene, we must divide the last term given in the sequence: 

For a problem like this, you can always use the answers to find your correct answer. By choosing each number, you can find the other two options and then add together your values. You would, for instance, take  and say, "The other two must be  and ." Then, adding them to get , you will know that this is not correct.

However, you can do this much more easily with algebra. You know that three consecutive integers are going to look like:

, where  is the price of the least expensive candy. Thus, you know that the total price of your candies can be represented in the following manner:

This simplifies to:

Solving for , you get:

Remember that you need to find the highest priced candy. Therefore, the answer is  or .

In this geometric sequence, the numbers get smaller; therefore, there will be a fractional ratio. We can determine the common ratio by dividing one of the terms by the term immediately preceding it. In the above sequence, use the smallest numbers, 4 and 16, in order to make the calculation the easiest.

Now, address which four of the five statements are false.

There is a common ratio, not a common difference, in a geometric sequence. Thus the answer choice that mentions "common difference" is incorrect.

The common ratio is , not 4. Thus, the common ratio answer choice is incorrect.

The fifth term is 1 in this sequence, not 2, which eliminates another answer choice.

Finally, the sum of the third and fourth terms is 20, and the sum of the fourth and fifth terms is 5; the only possible correct answer is "The seventh term is ."

If the common ratio is r, then the sequence can be rewritten as 3, 3r, , . We see then that , which gives us that r=2. Therefore, the missing terms are 6 and 12.

The common difference is equal to

Plugging our values into this equation we can find the common difference.

Therefore, in this case the common difference is .

The common difference is equal to

.

Plugging in the values from this problem we get,

Therefore, in this case the common difference is .

The common difference is equal to

.

Plugging in the values from this problem we get,

Therefore, in this case the common difference is .