Summations and Sequences - Algebra II

The 11th term means there are 10 gaps in between the first term and the 11th term. Each gap has a difference of +4, so the 11th term would be given by 10 * 4 + 1 = 41.

The first term is 1.

Each term after increases by +4.

The nth term will be equal to 1 + (n – 1)(4).

The 11th term will be 1 + (11 – 1)(4)

1 + (10)(4) = 1 + (40) = 41

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 .

A simple way to find the term of an arithmetic sequence is to use the formula .

Here, is the term you are trying to find, is the first term, and is the common difference. For this question, the common difference is .

The common difference between the terms is half that between the second and fourth terms - that is:

Subtract this common difference from the second term to get the first:

You can either calculate the vaules of and and subtract, or notice from the formula that each succesive number in the sequence is 3 larger than the previous

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.

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 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 a pattern in the sequence. You will notice that each time you move from one number to the very next one, it increases by 7. That is, the difference between one number and the next is 7. Therefore, we can add 7 to 36 and the result will be 43. Thus .

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 .

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.

Determine what kind of sequence you have, i.e. whether the sequence changes by a constant difference or a constant ratio. You can test this by looking at pairs of numbers, but this sequence has a constant difference (arithmetic sequence).

So the sequence advances by subtracting 16 each time. Apply this to the last given term.

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)

This is a sequence that features every other positive, odd integers. The missing number in this case is 9.

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.

An arithmetic sequence is one in which the common difference is added to one term to get the next term. Thus, if the first term is 3, we add 5 to get the second term, and continue in this manner.

Thus, the first four terms are:

This is an arithmetic sequence since the difference between consecutive terms is the same (). To find the th term of an arithmetic sequence, use the formula

,

where is the first term, is the number of terms, and is the difference between terms. In this case, is , is , and is .

This is an arithmetic series. The formula to find the th term is:

where is the difference between each term.

To find the 35th term substitute for and

This is an artithmetic series. The explicit formula for an arithmetic sequence is:

Where represents the term, and is the common difference.

In this instance . Therefore: