Summations and Sequences - Algebra 2

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

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 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 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

a) First we will start with creating the list of consecutive odd integers with the required relationships satisfied.

Start by calling the first of these five integers . The next odd integer will therefore be , and the third will be and so on... We can list them in a table to keep track:

Now we need to translate the conditions given in the text of the problem into a mathematical expression. Write it out piece-by-piece. First let's right out these two statements separately,

...the sum of the 2nd plus 3-times the first...

...the sum of the 3rd integer plus 2-times the 5th integer...

In the text of the problem the first quantity "is one less than" the other quantity. In other words, the quantity is one less than . Putting everything together we obtain:

Now solve for ,

Now we can use this solution to find the other four integers:

b) Now that we have the first 5 terms, , we can find an explicit formula for the arithmetic sequence. The general form of this is equation is,

Where is the common difference between adjacent terms and is the first term in the sequence. Because each adjacent term has a difference of , we have .

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 .

This series is neither geometric nor arithmetic.

A geometric sequences is multiplied by a common ratio () each term. An arithmetic series adds the same additional amount () to each term. This series does neither.

Mutiplicative and subtractive are not types of sequences.

Therefore, the answer is none of the other answers.

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 .

This series is neither geometric nor arithmetic.

A geometric sequences is multiplied by a common ratio () each term. An arithmetic series adds the same additional amount () to each term. This series does neither.

Mutiplicative and subtractive are not types of sequences.

Therefore, the answer is none of the other answers.

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.

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

Where represents the term, and is the common difference.

In this instance . Therefore:

This is a geometric series. The explicit formula for any geometric series is:

, where is the common ratio and is the number of terms.

In this instance and .

Substitute into the equation to find the 20th term:

The explicit formula for a geometric series is

In this problem

Therefore:

This series is geometric. The explicit formula for any geometric series is:

Where represents the term, is the first term, and is the common ratio.

In this series .

Therefore the formula to find the 15th term 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 .

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.