### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #31 : How To Find The Missing Part Of A List

An arithmetic sequence begins

What number replaces the square?

**Possible Answers:**

**Correct answer:**

Since this is an arithmetic sequence, each entry in the sequence is obtained by adding the same number to the previous entry - this number is

Let be the number in the square. Then

### Example Question #32 : How To Find The Missing Part Of A List

The first two terms of an arithmetic sequence are

Which of the following expressions is equivalent to the fifth term?

**Possible Answers:**

**Correct answer:**

An arithmetic sequence is formed by adding the same expression to each term to get the next term; this common difference is

.

To obtain the fifth term, add to the second term three times - equivalently, add three times this to the second term;

### Example Question #33 : How To Find The Missing Part Of A List

A geometric sequence begins

.

What number replaces the circle?

**Possible Answers:**

**Correct answer:**

Since this is a geometric sequence, each entry in the sequence is obtained by multiplying the previous entry by the same number - this number is

.

Now we can find the next three entries in the sequence:

This replaces the square.

replaces the triangle.

replaces the circle and is therefore the correct answer.

### Example Question #34 : How To Find The Missing Part Of A List

An arithmetic sequence begins

What number replaces the circle?

**Possible Answers:**

**Correct answer:**

Since this is an arithmetic sequence, each entry in the sequence is obtained by adding the same number to the previous entry - this number is

.

The next three entries in the sequence are computed as follows:

, which replaces the square

, which replaces the triangle

, which replaces the circle

### Example Question #35 : How To Find The Missing Part Of A List

A geometric sequence begins

What number replaces the square?

**Possible Answers:**

**Correct answer:**

Each term of a geometric sequence is obtained by multiplying the previous one by the same number (common ratio); this number is

.

Let be the number in the square.

### Example Question #36 : How To Find The Missing Part Of A List

The* Fibonacci sequence* is formed as follows:

For all integers ,

Which of the following is true of , the one-thousandth number in this sequence?

**Possible Answers:**

**Correct answer:**

To express , the one-thousandth term of the sequence, in terms of and alone, we note that, by definition of the sequence, each term, except for the first two, is equal to the sum of the previous two. Therefore,

Also

, and, substituting:

and

,

the correct choice.

### Example Question #37 : How To Find The Missing Part Of A List

The Fibonacci sequence is defined as follows:

For integers , .

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to determine which is greater from the information given.

(a) and (b) are equal

(a) is greater

(b) is greater

**Correct answer:**

(a) is greater

The Fibonacci sequence begins as follows:

This sequence is seen to be an increasing sequence. Therefore, each term is greater than its preceding term. In particular,

If we substitute 51 for in the rule of the sequence, we get

, so

This makes (a) greater.

### Example Question #38 : How To Find The Missing Part Of A List

Define a sequence as follows:

For all integers , .

Which of the following expressions is equal to ?

**Possible Answers:**

**Correct answer:**

Setting :

Similarly,

Substituting: