# SAT II Math II : Sequences

## Example Questions

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### Example Question #1 : Sequences

Evaluate:

The series diverges

The series diverges

Explanation:

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

### Example Question #1 : Sequences

Give the next term in this sequence:

_______________

Explanation:

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

.

### Example Question #3 : Sequences

Give the next term in this sequence:

__________

Explanation:

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:

### Example Question #1 : Sequences

Give the next term in this sequence:

_____________

Explanation:

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:

### Example Question #5 : Sequences

Give the next term in this sequence:

_____________

The correct answer is not among the other responses.

Explanation:

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

### Example Question #6 : Sequences

A geometric sequence begins as follows:

Give the next term of the sequence.

Explanation:

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:

### Example Question #7 : Sequences

A geometric sequence begins as follows:

Give the next term of the sequence.

Explanation:

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:

.

### Example Question #8 : Sequences

A geometric sequence begins as follows:

Give the next term of the sequence.

Explanation:

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 :

### Example Question #1 : Sequences

A geometric sequence begins as follows:

Express the next term of the sequence in simplest radical form.

Explanation:

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:

### Example Question #141 : Sat Subject Test In Math Ii

The first and second terms of a geometric sequence are  and , respectively. In simplest form, which of the following is its third term?