ACT Math : Arithmetic Sequences

Study concepts, example questions & explanations for ACT Math

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Example Questions

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Example Question #1 : Other Arithmetic Sequences

Whish of the following numbers completes the sequence 40,33,27,22,18,15,...?

Possible Answers:

14

13

7.5

12

16

Correct answer:

13

Explanation:

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.

Example Question #1 : Arithmetic Sequences

Which of the following completes the number sequence 4, 7, 11, 16, __________ ?

Possible Answers:

Correct answer:

Explanation:

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

Example Question #1 : Arithmetic Sequences

Find the 50th term in the following sequence.

Possible Answers:

Correct answer:

Explanation:

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.

Example Question #1 : How To Find The Answer To An Arithmetic Sequence

What is the next term in the sequence: 

Possible Answers:

Correct answer:

Explanation:

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 .

Example Question #1 : How To Find The Answer To An Arithmetic Sequence

The sum of four consecutive integers is 42. What is the value of the greatest number?

Possible Answers:

14

13

15

12

11

Correct answer:

12

Explanation:

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:

.

Example Question #1 : Arithmetic Sequences

If the first day of the year is a Monday, what is the 295th day?

Possible Answers:

Monday

Saturday

Tuesday

Wednesday

Correct answer:

Monday

Explanation:

The 295th day would be the day after the 42nd week has completed. 294 days/7 days a week = 42 weeks. The next day would therefore be a monday.

Example Question #1 : Nth Term Of An Arithmetic Sequence

If the first two terms of a sequence are \small \pi and \small 2\pi ^{2}, what is the 38th term?

Possible Answers:

\small 2^{37}\pi ^{38}

\small 2^{37}\pi ^{37}

\small 2\pi ^{38}

\small 2^{38}\pi ^{38}

\small 2\pi ^{37}

Correct answer:

\small 2^{37}\pi ^{38}

Explanation:

The sequence is multiplied by \small 2\pi each time.

Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

Find the  term of the following sequence:

Possible Answers:

Correct answer:

Explanation:

The formula for finding the  term of an arithmetic sequence is as follows:

where

= the difference between consecutive terms

= the number of terms

Therefore, to find the  term:

Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

What is the rd term of the following sequence:?

Possible Answers:

Correct answer:

Explanation:

Notice that between each of these numbers, there is a difference of ; however the first number is , the second , and so forth. This means that for each element, you know that the value must be , where  is that number's place in the sequence. Thus, for the rd element, you know that the value will be  or .

Example Question #22 : Sequences

What is the th term in the following series of numbers: ?

Possible Answers:

148

Correct answer:

Explanation:

Notice that between each of these numbers, there is a difference of . This means that for each element, you will add . The first element is  or . The second is  or , and so forth... Therefore, for the th element, the value will be  or .

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