Patterns - SSAT Upper Level: Quantitative

Card 1 of 100

0

Didn't Know

0

Didn't Know

Knew It

0

1 of 2019 left

Question

Which of the following numbers can complete the sequence?

Tap to reveal answer

Answer

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

For example, the number after 13 is 25 because:

$13\cdot2-1=26-1=25$

Thus, the number after 25 is equal to:

$25\cdot2-1=50-1=49$

← Didn't Know|Knew It →

Question

Define a sequence of numbers as follows:

$a_{1}= 4$

For all integers , $a_{n}= a_{n-1}+n^{2}$

Evaluate $a_{4}$ .

Tap to reveal answer

Answer

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.

$a_{n}= a_{n-1}+n^{2}$

$a_{2}= a_{2-1}+2^{2}$

$= a_{1}+4$

$a_{3}= a_{3-1}+3^{2}$

$= a_{2}+9$

$a_{4}= a_{4-1}+4^{2}$

$= a_{3}+16$

33 is the correct choice.

← Didn't Know|Knew It →

Question

Define a sequence of numbers as follows:

$a_{1}= 4$

For all integers , $a_{n}= \left (a_{n-1}+n \right )^{2}$

Evaluate $a_{3}$ .

Tap to reveal answer

Answer

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.

$a_{n}= \left (a_{n-1}+n \right )^{2}$

$a_{2}= \left (a_{2-1}+2 \right )^{2}$

$= \left (a_{1}+2 \right )^{2}$

$= \left (4+2 \right )^{2}$

$= 6^{2}$

$a_{3}= \left (a_{3-1}+3 \right )^{2}$

$= \left (a_{2}+3 \right )^{2}$

$= \left (36+3 \right )^{2}$

$= 39^{2}$

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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 $a_{1} = 6$ and the common ratio is $r = \frac{a_{2}}{a_{1}} = \frac{5}{6}$ ; therefore,

$S = \frac{6}{1- \frac{5}{6}} = \frac{6}{\frac{1}{6}} = 6 \cdot 6 = 36$

← Didn't Know|Knew It →

Question

Give the sum of the infinite geometric series that begins

Tap to reveal answer

Answer

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,

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

← Didn't Know|Knew It →

Question

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?

Tap to reveal answer

Answer

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:

← Didn't Know|Knew It →

Question

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?

Tap to reveal answer

Answer

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.

← Didn't Know|Knew It →

Question

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?

Tap to reveal answer

Answer

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:

The number inside the next arrow is

so the correct choice is Figure (b).

← Didn't Know|Knew It →

Question

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 ?

Tap to reveal answer

Answer

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,

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

← Didn't Know|Knew It →

Question

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 ?

Tap to reveal answer

Answer

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 sum,

.

← Didn't Know|Knew It →

Question

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 ?

Tap to reveal answer

Answer

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,

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

← Didn't Know|Knew It →

Question

A sequence of numbers begins:

What is the one-hundredth entry in this sequence?

Tap to reveal answer

Answer

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

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

← Didn't Know|Knew It →

Question

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

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

Tap to reveal answer

Answer

Replace with and with 1:

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

← Didn't Know|Knew It →

Question

Find the sum of this infinite geometric series:

$4 + \frac{7}{2} + \frac{49}{16} + ... + 4 \cdot \left ( \frac {7}{8} \right ) ^{n}+...$

Tap to reveal answer

Answer

The sum of an infinite series with first term and common ratio is

$S = \frac{a}{1-r}$

Set $a=4,r= \frac{7}{8}$ , and evaluate:

$S = \frac{a}{1-r} = \frac{4}{1-\frac{7}{8}} = \frac{4}{\left ( \frac{1}{8}\right )} = 4 \cdot 8=32$

← Didn't Know|Knew It →

Question

Give the next number in the following sequence:

Tap to reveal answer

Answer

The sequence is generated by alternately adding 2, then multiplying by 2:

$5 \times 2 = 10$

$12 \times 2 = 24$

$26 \times 2 = 52$

$54 \times 2 = \underline{108}$ , which is the correct choice.

← Didn't Know|Knew It →

Question

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

$6, 18, 10, 30, 22, 66, \square, \bigcirc...$

Tap to reveal answer

Answer

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

$6 \times 3 = 18$

$10 \times 3 = 30$

$22 \times 3 = 66$

- This number replaces the square.

$58 \times 3 = 174$ - This number replaces the circle.

← Didn't Know|Knew It →

Question

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

$20, 10, 16, 8, 14, 7, \square, \bigcirc...$

Tap to reveal answer

Answer

The sequence is generated by alternately dividing by 2 and adding 6:

$20 \div 2 = 10$

$16 \div 2 = 8$

$14 \div 2 = 7$

- This number replaces the square.

$13\div 2 = 6\frac{1}{2}$ - This number replaces the circle.

← Didn't Know|Knew It →

Question

Define an operation on the real numbers as follows:

$a \bigtriangleup b = 3a - 2b$

Find the value of that makes this statement true:

$x\bigtriangleup 5 = 32$

Tap to reveal answer

Answer

Replace in the defintiion:

$a \bigtriangleup b = 3a - 2b$

$x\bigtriangleup 5 =3x - 2 \cdot 5$

$x\bigtriangleup 5 =3x - 10$

Now, set this equal to 32 and solve for :

$x\bigtriangleup 5 = 32$

$3x \div 3= 42 \div 3$

← Didn't Know|Knew It →

Question

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

$1, 9, 25, 49, 81, 121, \bigcirc...$

Tap to reveal answer

Answer

The sequence comprises the squares of the odd integers, in order:

$1^{2} = 1$

$3^{2} = 9$

...

$11^{2} = 121$

The next number, which replaces the circle, is

$13^{2} = 169$ .

← Didn't Know|Knew It →

Question

Define an operation on the set of real numbers as follows:

$a; \S; b = a^{2} + b ^{2}$

Evaluate:

$-4; \S; \left ( -5 \right )$

Tap to reveal answer

Answer

$a; \S; b = a^{2} + b ^{2}$

$-4; \S; \left ( -5 \right ) = \left ( -4 \right )^{2} + \left ( -5\right ) ^{2}$

$= \left ( -4 \right )\times \left ( -4 \right ) + \left ( -5\right ) \times \left ( -5 \right )$

← Didn't Know|Knew It →

0

Knew It