The first six elements are as follows:

Add them:

Let the common ratio of the sequence be . Then The first three terms of the sequence are . The third term is 108, so

or .

The common ratio can be either - not enough information exists for us to determine which.

The sixth term is

If , the seventh term is .

If , the seventh term is .

Therefore, not enough information exists to determine the sixth term of the sequence.

Let the common ratio of the sequence be . Then The first three terms of the sequence are . The third term is 50, so

or .

Not enough information is given to choose which one is the common ratio. But the seventh term is

If , the seventh term is .

If , the seventh term is .

Either way, the seventh term is 31,250.

When you are dealing with arithmetic means, it is best to define one number in the sequence as x and every other number relative to x.

Because we are trying to find the largest of three numbers, let's define x as the largest number in the equation. Because each number is a consecutive odd number, we must subtract 2 to get to the next number in the sequence.

x: largest number in sequence

x-2: middle number in sequence

x-4: smallest number in sequence

Now, let's make an equation finds the sum of all the numbers in the sequence and set it equal to 354.

117 is the largest number in the sequence.

To check yourself, you can add up the numbers in the sequence {113, 115, 117}.

The first number is multiplied by three

.

Then it is divide by two

.

The following is multiplied by three

then divided by two

.

That makes the next step to multiply by three which gives us

.

Rewrite the first term in fraction form: .

The sequence now begins

,...

Rewrite the terms with their least common denominator, which is :

The common difference of the sequence is the difference of the second and first terms, which is

.

The rule for term of an arithmetic sequence, given first term and common difference , is

;

Setting , , and , we can find the tenth term by evaluating the expression:

,

the correct response.

Using the Product of Radicals principle, we can simplify the first two terms of the sequence as follows:

The common ratio of a geometric sequence is the quotient of the second term and the first:

Multiply the second term by the common ratio to obtain the third term:

The common ratio of a geometric sequence is the quotient of the second term and the first:

Simplify this common ratio by multiplying both numerator and denominator by :

Multiply the second term by the common ratio to obtain the third term:

Let be the common ratio of the geometric sequence. Then

and

Therefore,

,

and

Setting :

.

Substituting for and , either

.

The second term can be either or , the former of which is a choice.

Let be the common ratio of the geometric sequence. Then

and

Therefore,

,

Setting , and applying the Quotient of Radicals Rule:

Taking the square root of both sides:

Substituting, and applying the Product of Radicals Rule:

Since all elements of the sequence are positive, .

The common ratio of a geometric sequence is the quotient of the third term and the second:

Multiplying numerator and denominator by , this becomes

The second term of the sequence is equal to the first term multiplied by the common ratio:

.

so equivalently:

Substituting:

,

the correct response.

An infinite series converges to a sum if and only if . However, in the series , this is not the case, as . This series diverges.

The pattern becomes more clear if each term is rewritten as a single radical expression:

The th term is . The next (seventh) term is therefore

The key to finding the next term lies in the denominators of the third term onwards. They are terms of the Fibonacci sequence, which begin with the terms 1 and 1 and whose subsequent terms are each formed by adding the previous two.

The th term of the sequence is the number , where is the th number in the Fibonacci sequence (since the first two Fibonacci numbers are both 1, the first two terms being 0 fits this pattern). The Fibonacci number following 13 and 21 is their sum, 34, so the next number in the sequence is

.

Each term is derived from the next by adding a perfect square integer; the increment increases from one square to the next higher one each time. To maintain the pattern, add the next perfect square, 36:

Each term is derived from the previous term by doubling the latter and alternately adding and subtracting 1, as follows:

The next term is derived as follows:

The common ratio of a geometric sequence is the quotient of the second term and the first:

Multiply the second term by the common ratio to obtain the third term:

Rewrite the first term as a fraction:

The common ratio of a geometric sequence can be found by dividing the second term by the first, so

The third term is equal to the second term multiplied by this common ratio:

.

The common ratio of a geometric sequence is the quotient of the second term and the first:

Multiply this common ratio by the second term to get the third term:

This can be expressed in standard form by rationalizing the denominator; do this by multiplying numerator and denominator by the complex conjugate of the denominator, which is :

The common ratio of a geometric sequence is the quotient of the second term and the first. Using the Quotient of Radicals property, we can obtain:

Multiply the second term by the common ratio, then simplify using the Product Of Radicals Rule, to obtain the third term: