SSAT Upper Level Quantitative - Patterns

A sequence of numbers begins:

What is the one-hundredth entry in this sequence?

Explanation

Let be the $n\mathrm{th}$ entry in the sequence. Then $f(n) = n^{2} + 1$ . The one-hundredth entry is therefore

$f(100) = 100 ^{2}+1=10,001$

A sequence of numbers begins:

What is the one-hundredth entry in this sequence?

Explanation

Let be the $n\mathrm{th}$ entry in the sequence. Then $f(n) = n^{2} + 1$ . The one-hundredth entry is therefore

$f(100) = 100 ^{2}+1=10,001$

Crosses

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

What value replaces ?

$13\textup{,}530$

$41\textup{,}330$

Explanation

In each of the crosses, the lower left and lower right entries are the sum and product of the top two entries, respectively. The top two numbers of the next cross are the same as the bottom two of the current cross. Therefore, the fourth figure has as its top two entries 41 and 330 (the bottom two of the third figure). is equal to their product,

$41 \times 330 = 13\textup{,}530$ .

Define $a \bigtriangledown b = ab + b + 1$

Which of the following expressions is equal to $xy \bigtriangledown 1$ ?

Explanation

Replace with and with 1:

$xy \bigtriangledown 1 = xy \cdot 1 + 1 + 1 = xy + 2$

Triangles

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 ?

Explanation

The two numbers along the upper sides of each triangle are those in the previous triangle, increased by 1. Therefore,

and .

The number along the bottom side of each triangle is the product of the other two numbers. Therefore,

$c = a \cdot b = 7 \cdot8 = 56$ .

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?

Anya, Brian, Donna

Insufficient information is given to answer the question.

Anya, Clark, Donna

Anya, Brian, Clark

Brian, Donna, Clark.

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.

Give the sum of the infinite geometric series that begins

$3\frac{10}{13}$

$8\frac{1}{6}$

Explanation

The sum of an infinite geometric series with initial term $a_{1}$ and common ratio is:

$S = \frac{a_{1}}{1-r}$ .

The initial term is $a_{1} = 7$ and the common ratio is $r = \frac{a_{2}}{a_{1}} = \frac{-6}{7} = -\frac{6}{7}$ ; therefore,

$S = \frac{a_{1}}{1-r} = \frac{7}{1-\left ( - \frac{6}{7}\right )} =\frac{7}{\frac{13}{7}} = 7 \cdot \frac{7}{13} = \frac{49}{13} = 3\frac{10}{13}$

Divided difference

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:

Crosses

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

What value replaces ?

Explanation

Assorted 1

The top row in the above diagram shows a sequence of figures. Which figure in the bottom row is the next one in the sequence?

Figure (b)

Figure (a)

Figure (d)

Figure (c)

Explanation

Going from figure to figure, the arrow is rotating one quarter of a turn clockwise each time. Therefore, in the next figure, the arrow should be pointing to the right, thereby eliminating Figures (c) and (d) as choices and leaving Figures (a) and (b).

Starting with the third figure, each number is obtained by adding the numbers in the previous two figures (this is called the Fibonacci sequence), as follows: