Sequences - ACT Math

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 ."

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 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 .

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:

Sequencing problems require us to look at the numbers given to us ad decipher a pattern.

7 – 4 = 3 so 3 was added to the first number (4)

11 – 7 = 4 so 4 was added to the second number (7)

16 – 11 = 5 so 5 was added to the third number (11)

If it is to continue in this pattern, then 6 should be added to 16, yielding 22 as the correct answer

To get the next term in the sequence you subtract a decreasing amount from the preceding term. You subtract 7 from 40 to get 33, then 6 from 33 to get 27, and so on until you subtract 2 from 15 to get 13.

A sequence is simply a list of numbers that follow some sort of consistent rule in getting from one number in the list to the next one. Sequences generally fall into three categories: arithmetic, geometric, or neither.

In arithmetic sequences, I add the same number each time to get from one number to the next. In other words, the difference between any two consecutive numbers in my list is the same.

In geometric sequences, I multiply by the same number each time to get from one number to the next. In other words, the ratio between any two consecutive numbers in my list is the same.

Finally, sequences that are neither, still follow some rule, but it just happens not to be one of these two.

Looking at our sequence, we might quickly notice that each number is simply 7 more than the number before. In other words, I can find the next number by adding 7 each time. Hence, our sequence is arithmetic.

Unfortunately, we need to find the 50th term in this sequence, and the problem only got us through the first four. A simple (yet way too time-consuming approach) would be to keep adding 7 until we get to term number 50. Not only is that the long way, we also risk losing count and ending up on the wrong term. So what's the easier way?

The easier way hinges on the fact that I am simply adding 7 over and over again. If I want to find the 2nd term, I start with the 1st term and add 7 once.

To find the 3rd term, I add 7 twice.

You might already see the pattern. For the 4th term I would add 7 three times, for the 5th four times, 6th five times, etc.

Notice that to find any term, I simply add 7 one less time than the number of the term. Therefore, to find the 50th term, I would add 7 forty-nine times.

But adding 7 forty-nine times is the same as adding forty-nine 7s. But forty-nine 7s are the same as 49 times 7.

Therefore, to find the 50th term, I simply need to add 343 to our starting value.

Because it is an arithmetic sequence, the difference between each term is the same. Therefore find out the difference between any two consecutive terms. and so the sequence increases by 5 each term. Thus the answer is .

We can represent four consecutive integers using the following expressions:

Create an equation using the information in the problem.

Subtract 6 from both sides of the equation.

Divide both sides of the equation by 4.

The highest number in the series is the x-variable plus 3. We can write the following:

.

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.

Finding the common difference is fairly simple. We simply subtract the first term from the second. 7-3 = 4, so 4 is our common difference. So each term is going to be 4n plus something:

We know the first term is 3, so we can plug in that to our equation.

So the explicit form of our arithmetic sequence is

.

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 .

The sequence is defined by an = 4_n –_ 3 for such n = 1,2,3,4....

What is the next term in the following sequence?

This is an arithmetic sequence with a common difference of . To find the next term in an arithmetic sequence, add the common difference to the previously listed term:

This question can be answered by analyzing the sequence provided and determining the pattern. The first term is , and the second term is The third term is Thus, has been added to in order to obtain , and has been added to in order to obtain This shows that is added to each preceding term in the sequence in order to obtain the next term. The complete sequence from terms one through six is shown below.

Thus, the sixth term is