ACT Math : How to find consecutive integers

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : Sequences

Which of the following numbers completes the sequence 3, 10, 16, 21, 25, __?

Possible Answers:

28

30

29

31

32

Correct answer:

28

Explanation:

If you look at the pattern, you see that the numbers are steadily increasing. You have to figure out how much is being added to the previous number to produce the next number in the sequence. You see that the amount added is 7, then 6, then 5, then 4 and following the pattern would lead you to add three to the last number to get the missing value. 25+3 is 28, giving you your answer.

Example Question #1 : Sequences

If four consecutive odd integers greater than 9 are added together, what is the smallest possible sum of those four integers?

Possible Answers:

Correct answer:

Explanation:

The 4 consecutive of integers greater than 9 (but not including 9) are 11, 13, 15, 17.  Added together, we get 56.

Example Question #2 : How To Find Consecutive Integers

There are two consectutive positive integers  and , and their product is 132. 

What is the value of the larger integer?

Possible Answers:

Correct answer:

Explanation:

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

 

Example Question #93 : Integers

Five students are lined up by height so that their heights are ordered in a consecutive manner. If the sum of their heights (in inches) is  inches, what is the height of the second tallest student?

Possible Answers:

Correct answer:

Explanation:

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 .

Example Question #4 : Consecutive Integers

The sum of the squares of three consecutive odd integers is .

Which of the following is the smallest of of these three numbers?

Possible Answers:

Not able to be determined.

Correct answer:

Not able to be determined.

Explanation:

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.

Example Question #1 : How To Find Consecutive Integers

What is the next number in the geometric sequence?

Possible Answers:

Correct answer:

Explanation:

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: 

Example Question #6 : Consecutive Integers

The prices of three candies are consecutively priced. If the total price of the candies is , what is the cost of the highest priced candy?

Possible Answers:

Correct answer:

Explanation:

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 .

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