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SSAT Upper Level Math : How to find algebraic patterns

Study concepts, example questions & explanations for SSAT Upper Level Math

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All SSAT Upper Level Math Resources

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Example Questions

Example Question #1 : How To Find Algebraic Patterns

Find the sum of this infinite geometric series:

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

Possible Answers:

Correct answer:

Explanation:

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:

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

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Example Question #2 : How To Find Algebraic Patterns

Give the next number in the following sequence:

3, 5, 10, 12, 24, 26, 52, 54, _____

Possible Answers:

102

108

104

Correct answer:

108

Explanation:

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

3 + 2= 5

$5 \times 2 = 10$

10+2 = 12

$12 \times 2 = 24$

24 + 2 = 26

$26 \times 2 = 52$

52 + 2 = 54

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

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Example Question #3 : How To Find Algebraic Patterns

Define an operation on the real numbers as follows:

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

Find the value of that makes this statement true:

$x\bigtriangleup 5 = 32$

Possible Answers:

$x = \frac{2}{3}$

$x = 20 \frac{2}{3}$

$x = 7\frac{1}{3}$

x= 14

There is no such value of .

Correct answer:

x= 14

Explanation:

Replace a=x, b=5 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 - 10 = 32

3x - 10 + 10= 32+ 10

3x = 42

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

x= 14

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Example Question #4 : Algebra

What number replaces the circle?

$1, 1, 2, 4, 6, 18, 21, 84, 88, \square, \bigcirc...$

Possible Answers:

357

445

352

465

440

Correct answer:

445

Explanation:

The sequence is formed by alternately multiplying by a number, then adding the same number; the number incrementally increases every other term.

$1 \times 1 = 1$

1 + 1 = 2

$2 \times 2 = 4$

4 + 2 = 6

$6 \times 3 = 18$

18 + 3 = 21

$21 \times4 = 84$

84 + 4 = 88

$88 \times 5 = 440$ , the number which replaces the square.

440 + 5 = 445 , the number which replaces the circle.

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Example Question #4 : How To Find Algebraic Patterns

$\left \lfloor N \right \rfloor$ is defined as the greatest integer less than or equal to .

Evaluate the expression for x = 0.7 :

$\left \lfloor 2x +0.5 \right \rfloor + \left \lfloor 3x +0.2\right \rfloor$

Possible Answers:

5.2

4.2

Correct answer:

Explanation:

Substitute:

$\left \lfloor 2x +0.5 \right \rfloor + \left \lfloor 3x +0.2\right \rfloor$

$= \left \lfloor 2 \cdot 0.7 +0.5 \right \rfloor + \left \lfloor 3 \cdot 0.7 +0.2\right \rfloor$

$= \left \lfloor 1.4 +0.5 \right \rfloor + \left \lfloor 2.1 +0.2\right \rfloor$

$= \left \lfloor 1.9\right \rfloor + \left \lfloor 2.3\right \rfloor$

=1+2 = 3

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Example Question #5 : How To Find Algebraic Patterns

$\left \lceil N\right \rceil$ is defined as the least integer greater than or equal to .

Evaluate for x = 0.8 :

$\left \lceil 4 x -0.4 \right \rceil + 5x+ 0.3$

Possible Answers:

7.3

6.3

Correct answer:

7.3

Explanation:

Substitute x = 0.8 :

$\left \lceil 4 x -0.4 \right \rceil + 5x+ 0.3$

$= \left \lceil 4 \cdot 0.8 -0.4 \right \rceil + 5 \cdot 0.8 + 0.3$

$= \left \lceil 3.2 -0.4 \right \rceil + 4 + 0.3$

$= \left \lceil 2.8 \right \rceil + 4 + 0.3$

=3 + 4 + 0.3 = 7.3

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Example Question #6 : How To Find Algebraic Patterns

What number replaces the circle in this sequence?

$\bigcirc , \square , 8, 6, 12, 10, 20, 18, 36....$

Possible Answers:

Correct answer:

Explanation:

This sequence is alternately generated by subtracting two and multiplying by two.

We can therefore find the numbers that replace the square and the circle by reversing the pattern - alternately dividing by two and adding two beginning at 36:

$36 \div 2 = 18$

18 + 2 = 20

$20 \div 2 =10$

10 + 2 = 12

$12 \div 2 = 6$

6 + 2 = 8

$8 \div 2 = 4$ - this replaces the circle

4+2=6 - this replaces the square

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Example Question #7 : Algebra

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

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

Possible Answers:

196

161

141

144

169

Correct answer:

169

Explanation:

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

$13^{2} = 169$ .

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Example Question #7 : How To Find Algebraic Patterns

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

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

Evaluate:

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

Possible Answers:

The expression is undefined.

Correct answer:

Explanation:

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

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

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

= 16 +25 = 41

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Example Question #8 : How To Find Algebraic Patterns

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

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

Possible Answers:

$11\frac{1}{2}$

$6\frac{1}{2}$

$7\frac{1}{2}$

$10\frac{1}{2}$

$9\frac{1}{2}$

Correct answer:

$6\frac{1}{2}$

Explanation:

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

$20 \div 2 = 10$

10 + 6 = 16

$16 \div 2 = 8$

8 + 6 = 14

$14 \div 2 = 7$

7 + 6 = 13 - This number replaces the square.

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

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SSAT Upper Level Math : How to find algebraic patterns

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #1 : How To Find Algebraic Patterns

Example Question #2 : How To Find Algebraic Patterns

Example Question #3 : How To Find Algebraic Patterns

Example Question #4 : Algebra

Example Question #4 : How To Find Algebraic Patterns

Example Question #5 : How To Find Algebraic Patterns

Example Question #6 : How To Find Algebraic Patterns

Example Question #7 : Algebra

Example Question #7 : How To Find Algebraic Patterns

Example Question #8 : How To Find Algebraic Patterns

All SSAT Upper Level Math Resources

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