Sequences - Algebra

If is one odd number, then the next odd number is . If their product is 143, then the following equation is true.

Distribute into the parenthesis.

Subtract 143 from both sides.

This can be solved by factoring, or by the quadratic equation. We will use the latter.

We are told that both integers are positive, so .

The other integer is .

Know that the general rule for an arithmetic sequence is

,

where represents the first number in the sequence, is the common difference between consecutive numbers, and is the -th number in the sequence.

In our problem, .

Each time we move up from one number to the next, the sequence increases by 3. Therefore, .

The rule for this sequence is therefore .

The first term for the sequence is where . Thus,

So the first term is 4. Repeat the same thing for the second , third , fourth , and fifth terms.

We see that the first five terms in the sequence are

When finding consecutive numbers assign the first number a variable.

If the first number is assigned the letter n, then the second number that is consecutive must be and the third number must be .

Write it out as an equation and it should look like:

Simplify the equation then,

If then

And

So the answer is

Let the first number be .

If is an odd number, the next odd numbers will be:

, , , and

The fourth highest number would then be:

Set up an equation where the sum of all these numbers add up to .

Simplify this equation.

Subtract 20 from both sides.

Simplify both sides.

Divide by five on both sides.

Corresponding to the five numbers, the set of five consecutive numbers that add up to are:

The fourth largest number would be .

In each case, the terms increase by the same number, so all of these sequences are arithmetic.

Each term is the result of adding 1 to the previous term. 1 is the common difference.

Each term is the result of subtracting 1 from - or, equivalently, adding to - the previous term. is the common difference.

The common difference is 0 in a constant sequence such as this.

Each term is the result of adding to the previous term. is the common difference.

First, find the common difference for the sequence. Subtract the first term from the second term.

Subtract the second term from the third term.

To find the next value, add to the last given number.

In an arithmetic sequence the amount that the sequence grows or shrinks by on each successive term is the common difference. This is a fixed number you can get by subtracting the first term from the second.

So the sequence is adding 12 each time. Add 12 to 25 to get the third term.

So the unknown term is 37. To double check add 12 again to 37 and it should equal the fourth term, 49, which it does.

In each group of numbers, compare the difference of the second and first terms to that of the third and second terms. The group in which they are unequal is the correct choice.

The last group of numbers is the correct choice.

In arithmetic sequences, the common difference is simply the value that is added to each term to produce the next term of the sequence. When solving this equation, one approach involves substituting 5 for to find the numbers that make up this sequence. For example,

so 14 is the first term of the sequence. However, a much easier approach involves only the last two terms, and .

The difference between these expressions is 8, so this must be the common difference between consecutive terms in the sequence.

An arithmetic sequence adds or subtracts a fixed amount (the common difference) to get the next term in the sequence. If you know you have an arithmetic sequence, subtract the first term from the second term to find the common difference.

An arithmetic sequence adds or subtracts a fixed amount (the common difference) to get the next term in the sequence. If you know you have an arithmetic sequence, subtract the first term from the second term to find the common difference.

(i.e. the sequence advances by subtracting 27)

The common difference is the distance between each number in the sequence. Notice that each number is 3 away from the previous number.

What is the common difference in the following sequence?

Common differences are associated with arithematic sequences.

A common difference is the difference between consecutive numbers in an arithematic sequence. To find it, simply subtract the first term from the second term, or the second from the third, or so on...

See how each time we are adding 8 to get to the next term? This means our common difference is 8.

The common difference in this set is the linear amount spaced between each number in the set.

Subtract the first number from the second number.

Check this number by subtracting the second number from the third number.

Each spacing, or common difference is:

The common difference can be determined by subtracting the first term with the second term, second term with the third term, and so forth. The common difference must be similar between each term.

The distance between the first and second term is .

The distance between the second and third term is .

The distance between the third and fourth term is .

The fractions may seem as though they have a common difference since the denominators are increasing by one for each term, but there is no common difference among the numbers.

The answer is:

In order to determine the common difference, subtract the first term from the second term.

Verify that this is the same for the difference of the third and second terms.

The set of data is increasing at increments of five.

The common difference is:

The question states that the sequence is arithmetic, which means we find the next number in the sequence by adding (or subtracting) a constant term. We know two of the values, separated by one unknown value.

We know that is equally far from -1 and from 13; therefore is equal to half the distance between these two values. The distance between them can be found by adding the absolute values.

The constant in the sequence is 7. From there we can go forward or backward to find out that .

By taking the difference between two adjacent numbers in the sequence, we can see that the common difference increases by one each time.

Our next term will fit the equation , meaning that the next term must be .

After , the next term will be , meaning that the next term must be .

Finally, after , the next term will be , meaning that the next term must be

The question asks for the sum of the next three terms, so now we need to add them together.

The two things we need to find out are HOW the sequence changes (adddition, subtraction, multiplication, division, etc.) and by WHAT factor.

Start by finding the difference between the first two terms.

Now let's find the difference between the 2nd and 3rd given term.

Based on these two points, we can infer that this sequence changes by adding 13 to the previous term. Therefore...

the next term in the sequence is 22.