# SSAT Upper Level Math : How to find algebraic patterns

## Example Questions

### Example Question #11 : How To Find Algebraic Patterns

In the following numeric sequence, what number goes in place of the circle?

Explanation:

The sequence is generated by alternately multiplying by 3, then adding 8:

- This number replaces the square.

- This number replaces the circle.

### Example Question #11 : How To Find Algebraic Patterns

Anya, Brian, Clark, and Donna represented Central High in a math contest. The team score is the sum of the three highest scores; Anya outscored Clark, Brian outscored Donna; Donna outscored Clark. Whose scores were added to determine the team score?

Brian, Donna, Clark.

Anya, Brian, Clark

Anya, Brian, Donna

Anya, Clark, Donna

Insufficient information is given to answer the question.

Anya, Brian, Donna

Explanation:

Let  be the scores by Anya, Brian, Clark, and Donna, respectively.

Three inequalities can be deduced from these statements:

Anya outscored Clark:

Brian outscored Donna:

Donna outscored Clark:

The first and third statements can be combined to arrive at:

so Brian and Donna both outscored Clark. Since Anya outscored Clark also, Clark finished last among the four, and the team score was the sum of Anya's, Brian's and Donna's scores.

### Example Question #11 : Algebra

Which of the following numbers can complete the sequence?

Explanation:

Each subsequent number in this set is half the previous number, minus 1.

For example, the number after 13 is 25 because:

Thus, the number after 25 is equal to:

### Example Question #11 : Algebra

Give the sum of the infinite geometric series whose first two terms are 6 and 5, in that order.

Explanation:

The sum of an infinite geometric series with initial term  and common ratio  is:

.

The initial term is  and the common ratio is ; therefore,

### Example Question #11 : How To Find Algebraic Patterns

Examine the above figure. In the top row, the cubes of the whole numbers are written in ascending order. In each successive row, each entry is the difference of the two entries above it - five of those entries have been calculated for you.

What is the fifth entry in the third row?

Explanation:

The fifth entry in the third row is the difference of the sixth and fifth entries in the second row.

The sixth entry in the second row is the difference of 216 and 125:

The fifth entry in the second row is the difference of 343 and 216:

Now subtract these two:

### Example Question #11 : Ssat Upper Level Quantitative (Math)

Examine the above figure. In the top row, the cubes of the whole numbers are written in ascending order. In each successive row, each entry is the difference of the two entries above it - five of those entries have been calculated for you.

Suppose we were to extend the figure infinitely. What would be the tenth entry in the fourth row?

Explanation:

We do not actually need to calculate this entry; we can actually see the pattern if we just calculate the first few entries in the fourth row.

The fourth row is comprised entirely of 6's, so the tenth entry - the correct response - is 6.

### Example Question #11 : How To Find Algebraic Patterns

Give the sum of the infinite geometric series that begins

Explanation:

The sum of an infinite geometric series with initial term  and common ratio  is:

.

The initial term is  and the common ratio is ; therefore,

### Example Question #12 : Algebra

Define a sequence of numbers as follows:

For all integers

Evaluate .

Explanation:

Using the definition of this particular sequence we will plug in  to find,  then use that to find the next term and so one and so forth.

33 is the correct choice.

### Example Question #11 : Patterns

Define a sequence of numbers as follows:

For all integers

Evaluate .

Explanation:

Using the definition of this particular sequence we will plug in  to find,  then use that to find the next term and so one and so forth.

### Example Question #11 : How To Find Algebraic Patterns

The above diagram shows a sequence of figures. In the fourth figure, each of the three variables, , and , is replaced by a value.

What value replaces ?