Sequences - ACT Math

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 .

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

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:

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.

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.