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SSAT Upper Level Math : Arithmetic Sequences

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Example Questions

Example Question #1 : Other Arithmetic Sequences

The angle measures of a pentagon form an arithmetic sequence. The smallest angle measures $70^{\circ }$ $\displaystyle 70^{\circ }$. What does the largest angle measure?

Possible Answers:

$108^{\circ }$ $\displaystyle 108^{\circ }$

$142^{\circ }$ $\displaystyle 142^{\circ }$

$146^{\circ }$ $\displaystyle 146^{\circ }$

$138^{\circ }$ $\displaystyle 138^{\circ }$

$127^{\circ }$ $\displaystyle 127^{\circ }$

Correct answer:

$146^{\circ }$ $\displaystyle 146^{\circ }$

Explanation:

The measures of the five angles of a pentagon add up to $\displaystyle 180^o (5-2) = 540^o$, based on the formula $\displaystyle 180^o(n-2)$.

If the measures of the five angles form an arithmetic sequence, the angles have measures increasing by a some common difference, $\displaystyle d$.

$\displaystyle 70,70+d,70+2d,70+3d,70+4d$

Given this pattern and the total sum, we can solve for the common difference.

$70+ \left( 70+d\right )+\left( 70+2d\right )+\left( 70+3d\right )+\left( 70+4d\right ) = 540$ $\displaystyle 70+ \left( 70+d\right )+\left( 70+2d\right )+\left( 70+3d\right )+\left( 70+4d\right ) = 540$

$\displaystyle 350+10d = 540$

$\displaystyle 10d = 190$

d=19 $\displaystyle d=19$

The greatest of the angle measures is given by 70+4d $\displaystyle 70+4d$.

$70+4d = 70 + (4 )(19) = 146^{\circ }$ $\displaystyle 70+4d = 70 + (4 )(19) = 146^{\circ }$

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Example Question #2 : Arithmetic Sequences

Hunter makes $\$4.00$ $\displaystyle \$4.00$ for the first hour of work, $\$6.00$ $\displaystyle \$6.00$ for his second hour of work, $\$8.00$ $\displaystyle \$8.00$ for his third hour of work, and so on. How much money will he make if he works for $\displaystyle 11$ hours?

Possible Answers:

$\$24$ $\displaystyle \$24$

$\$130$ $\displaystyle \$130$

$\$180$ $\displaystyle \$180$

$\$154$ $\displaystyle \$154$

Correct answer:

$\$154$ $\displaystyle \$154$

Explanation:

The question is basically asking you to find the sum of the first $\displaystyle 11$ terms of the following arithmetic sequence:

$\displaystyle 4, 6, 8...$

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 11th 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=4 $\displaystyle a_1=4$

n=11 $\displaystyle n=11$

d=6-4=2 $\displaystyle d=6-4=2$

Now, plug in the information to find the value of the 11th term.

$y_{11}=4+(11-1)(2)=4+(10)(2)=4+20=24$ $\displaystyle y_{11}=4+(11-1)(2)=4+(10)(2)=4+20=24$

Now that we know the 11th term of the sequence, we can plug in that value into the equation for the sum to find what these first 11 terms add up to.

$\text{Sum}=\frac{11(4+24)}{2}=\frac{11(28)}{2}=154$ $\displaystyle \text{Sum}=\frac{11(4+24)}{2}=\frac{11(28)}{2}=154$

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Example Question #3 : Arithmetic Sequences

Find the $8^{th}$ $\displaystyle 8^{th}$ term of the arithmetic sequence:

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

Possible Answers:

$\displaystyle 28$

$\displaystyle 36$

$\displaystyle 34$

$\displaystyle 40$

Correct answer:

$\displaystyle 34$

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=-8 $\displaystyle a_1=-8$

n=8 $\displaystyle n=8$

$\displaystyle d=-2-(-8)=6$

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

$\displaystyle y_8=-8+(8-1)6=-8+(7)6=-8+42=34$

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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 : Arithmetic Sequences

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

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

Possible Answers:

$\displaystyle 69$

$\displaystyle 73$

$\displaystyle 81$

$\displaystyle 77$

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 #8 : 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:

$\displaystyle 56$

$\displaystyle 54$

$\displaystyle 58$

$\displaystyle 60$

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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All SSAT Upper Level Math Resources

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SSAT Upper Level Math : Arithmetic Sequences

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

All SSAT Upper Level Math Resources

Example Questions

Example Question #1 : Other Arithmetic Sequences

Example Question #2 : Arithmetic Sequences

Example Question #3 : Arithmetic Sequences

Example Question #2 : Arithmetic Sequences

Example Question #1 : 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 #8 : Arithmetic Sequences

Example Question #9 : Arithmetic Sequences

All SSAT Upper Level Math Resources

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