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SSAT Upper Level Math : Number Concepts and Operations

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

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All SSAT Upper Level Math Resources

6 Diagnostic Tests 311 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2 : Arithmetic Sequences

Find the $16^{th}$ $\displaystyle 16^{th}$ term of the following arithmetic sequence:

$\displaystyle 100, 94, 88, 82...$

Possible Answers:

$\displaystyle 16$

$\displaystyle 10$

$\displaystyle 4$

$\displaystyle 22$

Correct answer:

$\displaystyle 10$

Explanation:

To find any term in an arithmetic sequence, use the following formula:

$\displaystyle y_n=a_1+(n-1)d$

$\displaystyle y_n$ is the term we want to find
$\displaystyle a_1$ is the first term of the sequence
$\displaystyle n$ is the number of the term we want to find
$\displaystyle d$ is the common difference

For this sequence,

a_1=100 $\displaystyle a_1=100$

n=16 $\displaystyle n=16$

$\displaystyle d=94-100=-6$

Now, plug in the information to find the value of the 16th term.

$y_{16}=100+(-6)(16-1)=100+(-6)(15)=100-90=10$ $\displaystyle y_{16}=100+(-6)(16-1)=100+(-6)(15)=100-90=10$

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Example Question #1 : Other Arithmetic Sequences

Find the $9^{th}$ $\displaystyle 9^{th}$ term of the following arithmetic sequence:

$\displaystyle 45, 49, 53, 57...$

Possible Answers:

$\displaystyle 81$

$\displaystyle 77$

$\displaystyle 73$

$\displaystyle 69$

Correct answer:

$\displaystyle 77$

Explanation:

To find any term in an arithmetic sequence, use the following formula:

$\displaystyle y_n=a_1+(n-1)d$

$\displaystyle y_n$ is the term we want to find
$\displaystyle a_1$ is the first term of the sequence
$\displaystyle n$ is the number of the term we want to find
$\displaystyle d$ is the common difference

For this sequence,

a_1=45 $\displaystyle a_1=45$

n=9 $\displaystyle n=9$

$\displaystyle d=49-45=4$

Now, plug in the information to find the value of the 9th term.

$\displaystyle y_9=45+(4)(9-1)=45+(4)(8)=45+32=77$

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Example Question #1 : How To Find The Answer To An Arithmetic Sequence

Find the $12^{th}$ $\displaystyle 12^{th}$ term of the following arithmetic sequence:

$\displaystyle -1, 4, 9, 14...$

Possible Answers:

$\displaystyle 48$

$\displaystyle 66$

$\displaystyle 60$

$\displaystyle 54$

Correct answer:

$\displaystyle 54$

Explanation:

To find any term in an arithmetic sequence, use the following formula:

$\displaystyle y_n=a_1+(n-1)d$

$\displaystyle y_n$ is the term we want to find
$\displaystyle a_1$ is the first term of the sequence
$\displaystyle n$ is the number of the term we want to find
$\displaystyle d$ is the common difference

For this sequence,

a_1=-1 $\displaystyle a_1=-1$

n=12 $\displaystyle n=12$

$\displaystyle d=-4-(-1)=5$

Now, plug in the information to find the value of the 12th term.

$y_{12}=-1+(12-1)(5)=-1+(11)(5)=-1+55=54$ $\displaystyle y_{12}=-1+(12-1)(5)=-1+(11)(5)=-1+55=54$

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Example Question #6 : Arithmetic Sequences

Find the $7^{th}$ $\displaystyle 7^{th}$ term for the following arithmetic sequence:

$\displaystyle 18, 11, 4, -3...$

Possible Answers:

$\displaystyle -31$

$\displaystyle -10$

$\displaystyle -17$

$\displaystyle -24$

Correct answer:

$\displaystyle -24$

Explanation:

To find any term in an arithmetic sequence, use the following formula:

$\displaystyle y_n=a_1+(n-1)d$

$\displaystyle y_n$ is the term we want to find
$\displaystyle a_1$ is the first term of the sequence
$\displaystyle n$ is the number of the term we want to find
$\displaystyle d$ is the common difference

For this sequence,

a_1=18 $\displaystyle a_1=18$

n=7 $\displaystyle n=7$

$\displaystyle d=11-18=-7$

Now, plug in the information to find the value of the 7th term.

$\displaystyle y_7=18+(7-1)(-7)=18+(6)(-7)=18-42=-24$

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Example Question #7 : Arithmetic Sequences

Find the $10^{th}$ $\displaystyle 10^{th}$ term of the following arithmetic sequence:

$\displaystyle -15, -7, 1, 9...$

Possible Answers:

$\displaystyle 51$

$\displaystyle 65$

$\displaystyle 49$

$\displaystyle 57$

Correct answer:

$\displaystyle 57$

Explanation:

To find any term in an arithmetic sequence, use the following formula:

$\displaystyle y_n=a_1+(n-1)d$

$\displaystyle y_n$ is the term we want to find
$\displaystyle a_1$ is the first term of the sequence
$\displaystyle n$ is the number of the term we want to find
$\displaystyle d$ is the common difference

For this sequence,

a_1=-15 $\displaystyle a_1=-15$

n=10 $\displaystyle n=10$

$\displaystyle d=-7-(-15)=8$

Now, plug in the information to find the value of the 10th term.

$y_{10}=-15+(8)(10-1)=-15+(8)(9)=-15+72=57$ $\displaystyle y_{10}=-15+(8)(10-1)=-15+(8)(9)=-15+72=57$

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Example Question #2 : Other Arithmetic Sequences

Brandon is improving very quickly at math. He improves at a rate of $\displaystyle 2$ points per test. If he scored a $\displaystyle 42$ on his first test, $\displaystyle 44$ on his second test, and $\displaystyle 46$ on his third test, what score will he get on his $8^{th}$ $\displaystyle 8^{th}$ test?

Possible Answers:

$\displaystyle 56$

$\displaystyle 54$

$\displaystyle 60$

$\displaystyle 58$

Correct answer:

$\displaystyle 56$

Explanation:

You should recognize this as an arithmetic sequence:

$\displaystyle 42, 44, 46...$

The question is asking you to find the 8th term in this particular sequence.

To find any term in an arithmetic sequence, use the following formula:

$\displaystyle y_n=a_1+(n-1)d$

$\displaystyle y_n$ is the term we want to find
$\displaystyle a_1$ is the first term of the sequence
$\displaystyle n$ is the number of the term we want to find
$\displaystyle d$ is the common difference

Using the information given from the question,

a_1=42 $\displaystyle a_1=42$

n=8 $\displaystyle n=8$

d=2 $\displaystyle d=2$

Now, plug in the information to find the value of the 8th term.

$\displaystyle y_8=42+(8-1)(2)=42+(7)(2)=42+14=56$

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Example Question #9 : Arithmetic Sequences

Julia gets better every time she plays basketball. In her first game, she scored $\displaystyle 8$ points. In her second game, she scored $\displaystyle 12$ points, and in her third game, she scored $\displaystyle 16$ points. If she continues to improve her basketball skills at this same pace, how many points should she be scoring by her $12^{th}$ $\displaystyle 12^{th}$ game?

Possible Answers:

$\displaystyle 56$

$\displaystyle 44$

$\displaystyle 52$

$\displaystyle 48$

Correct answer:

$\displaystyle 52$

Explanation:

You should recognize this as an arithmetic sequence:

$\displaystyle 8, 12, 16...$

The question is asking you to find the 12th term in this particular sequence.

To find any term in an arithmetic sequence, use the following formula:

$\displaystyle y_n=a_1+(n-1)d$

$\displaystyle y_n$ is the term we want to find
$\displaystyle a_1$ is the first term of the sequence
$\displaystyle n$ is the number of the term we want to find
$\displaystyle d$ is the common difference

Using the information given from the question,

a_1=8 $\displaystyle a_1=8$

n=12 $\displaystyle n=12$

d=4 $\displaystyle d=4$

Now, plug in the information to find the value of the 12th term.

$y_{12}=8+(12-1)(4)=8+(11)(4)=8+44=52$ $\displaystyle y_{12}=8+(12-1)(4)=8+(11)(4)=8+44=52$

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Example Question #31 : Sequences And Series

Leon makes $\$2.00$ $\displaystyle \$2.00$ for the first hour of work, $\$6.00$ $\displaystyle \$6.00$ for his second hour of work, $\$10.00$ $\displaystyle \$10.00$ for his third hour of work, and so on. How much will he make for his $12^{th}$ $\displaystyle 12^{th}$ hour of work?

Possible Answers:

$\$46$ $\displaystyle \$46$

$\$42$ $\displaystyle \$42$

$\$50$ $\displaystyle \$50$

$\$54$ $\displaystyle \$54$

Correct answer:

$\$46$ $\displaystyle \$46$

Explanation:

You should recognize this as an arithmetic sequence:

$\displaystyle 2, 6, 10...$

The question is asking you to find the 12th term in this particular sequence.

To find any term in an arithmetic sequence, use the following formula:

$\displaystyle y_n=a_1+(n-1)d$

$\displaystyle y_n$ is the term we want to find
$\displaystyle a_1$ is the first term of the sequence
$\displaystyle n$ is the number of the term we want to find
$\displaystyle d$ is the common difference

Using the information given from the question,

a_1=2 $\displaystyle a_1=2$

n=12 $\displaystyle n=12$

d=4 $\displaystyle d=4$

Now, plug in the information to find the value of the 12th term.

$y_{12}=2+(12-1)(4)=2+(11)(4)=2+44=46$ $\displaystyle y_{12}=2+(12-1)(4)=2+(11)(4)=2+44=46$

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Example Question #12 : Other Arithmetic Sequences

Find the sum of the first $\displaystyle 8$ terms of the following arithmetic sequence:

$\displaystyle -10, -4, 2, 8...$

Possible Answers:

$\displaystyle 88$

$\displaystyle 70$

$\displaystyle 32$

$\displaystyle 80$

Correct answer:

$\displaystyle 88$

Explanation:

To find the sum of a certain number of terms in an arithmetic sequence, use the following formula:

$\text{Sum}=\frac{n(y_1+y_n)}{2}$ $\displaystyle \text{Sum}=\frac{n(y_1+y_n)}{2}$

$\displaystyle n=$ the number of the terms you have
$\displaystyle y_1=$ the first term of the sequence
$y_n=n^{th}$ $\displaystyle y_n=n^{th}$ term of the sequence

To find the sum, we need to first find the 8th term of the sequence.

To find any term in an arithmetic sequence, use the following formula:

$\displaystyle y_n=a_1+(n-1)d$

$\displaystyle y_n$ is the term we want to find
$\displaystyle a_1$ is the first term of the sequence
$\displaystyle n$ is the number of the term we want to find
$\displaystyle d$ is the common difference

Using the information given from the question,

a_1=-10 $\displaystyle a_1=-10$

n=8 $\displaystyle n=8$

$\displaystyle d=-4-(-10)=6$

Now, plug in the information to find the value of the 8th term.

$\displaystyle y_8=-10+(8-1)(6)=-10+(7)(6)=-10+42=32$

Now that we know the 8th term of the sequence, we can plug in that value into the equation for the sum to find what these first 8 terms add up to.

$\text{Sum}=\frac{8(-10+32)}{2}=\frac{(8)(22)}{2}=88$ $\displaystyle \text{Sum}=\frac{8(-10+32)}{2}=\frac{(8)(22)}{2}=88$

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Example Question #13 : Other Arithmetic Sequences

Find the sum of the first $\displaystyle 9$ terms of the following arithmetic sequence:

$\displaystyle 15, 11, 7, 3...$

Possible Answers:

$\displaystyle -27$

$\displaystyle -17$

$\displaystyle -9$

$\displaystyle -18$

Correct answer:

$\displaystyle -9$

Explanation:

To find the sum of a certain number of terms in an arithmetic sequence, use the following formula:

$\text{Sum}=\frac{n(y_1+y_n)}{2}$ $\displaystyle \text{Sum}=\frac{n(y_1+y_n)}{2}$

$\displaystyle n=$ the number of the terms you have
$\displaystyle y_1=$ the first term of the sequence
$y_n=n^{th}$ $\displaystyle y_n=n^{th}$ term of the sequence

To find the sum, we need to first find the 9th term of the sequence.

To find any term in an arithmetic sequence, use the following formula:

$\displaystyle y_n=a_1+(n-1)d$

$\displaystyle y_n$ is the term we want to find
$\displaystyle a_1$ is the first term of the sequence
$\displaystyle n$ is the number of the term we want to find
$\displaystyle d$ is the common difference

Using the information given from the question,

a_1=15 $\displaystyle a_1=15$

n=9 $\displaystyle n=9$

$\displaystyle d=11-15=-4$

Now, plug in the information to find the value of the 9th term.

$\displaystyle y_9=15+(9-1)(-4)=15+(8)(-4)=15+(-32)=17$

Now that we know the 9th term of the sequence, we can plug in that value into the equation for the sum to find what these first 9 terms add up to.

$\text{Sum}=\frac{9(15+(-17))}{2}=\frac{(9)(-2)}{2}=-9$ $\displaystyle \text{Sum}=\frac{9(15+(-17))}{2}=\frac{(9)(-2)}{2}=-9$

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All SSAT Upper Level Math Resources

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SSAT Upper Level Math : Number Concepts and Operations

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

All SSAT Upper Level Math Resources

Example Questions

Example Question #2 : Arithmetic Sequences

Example Question #1 : Other Arithmetic Sequences

Example Question #1 : How To Find The Answer To An Arithmetic Sequence

Example Question #6 : Arithmetic Sequences

Example Question #7 : Arithmetic Sequences

Example Question #2 : Other Arithmetic Sequences

Example Question #9 : Arithmetic Sequences

Example Question #31 : Sequences And Series

Example Question #12 : Other Arithmetic Sequences

Example Question #13 : Other Arithmetic Sequences

All SSAT Upper Level Math Resources

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