Arithmetic Sequences - ACT Math

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:

.

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

Begin by looking at the transitions from number to number in this series:

From to : Add

From to : Subtract

From to : Add

From to : Subtract

From what you can tell, you can guess that the next step will be to add . Thus, the next value will be .

This sequence is a little tricky. Notice that the second element is equal to the first. After that, the third is equal to the sum of the first two , next the fourth is equal to the sum of the second and the third , and the same continues for each element after this. Thus, the next element in the series will be equal to or .

The difference between each term is constant, thus the sequence is an arithmetic sequence.

Simply find the difference between each term, and add it to the last term to find the next term.

Use the formula _a_n = _a_1 + (n – 1)d

_a_6 = a_1 + 5_d

_a_9 = a_1 + 8_d

Subtracting these equations yields

_a_6 – a_9 = –3_d

–7 – 8 = –3_d_

d = 5

_a_1 = 33

Then use the formula for the series; = –30

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.

The sequence is multiplied by each time.

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:

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 .

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 .