How to find the answer to an arithmetic sequence - SSAT Upper Level Quantitative

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Question

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

Answer

The measures of the five angles of a pentagon add up to , based on the formula .

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

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

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

The greatest of the angle measures is given by .

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

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Question

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

Answer

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

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

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

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Question

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

Answer

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

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

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

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

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Question

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

Answer

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

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

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

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Question

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

Answer

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

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

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

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

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Question

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

Answer

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

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

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

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Question

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

Answer

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

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

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

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

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Question

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

Answer

You should recognize this as an arithmetic sequence:

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:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

Using the information given from the question,

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

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Question

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

Answer

You should recognize this as an arithmetic sequence:

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:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

Using the information given from the question,

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

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

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Question

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

Answer

You should recognize this as an arithmetic sequence:

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:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

Using the information given from the question,

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

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

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Question

Find the sum of the first terms of the following arithmetic sequence:

Answer

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

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

the number of the terms you have

the first term of the sequence

$y_n=n^{th}$ 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:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

Using the information given from the question,

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

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.

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

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Question

Find the sum of the first terms of the following arithmetic sequence:

Answer

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

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

the number of the terms you have

the first term of the sequence

$y_n=n^{th}$ 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:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

Using the information given from the question,

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

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.

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

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Question

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

Answer

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

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

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

the number of the terms you have

the first term of the sequence

$y_n=n^{th}$ 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:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

Using the information given from the question,

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

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

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.

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

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Question

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

Answer

The measures of the five angles of a pentagon add up to , based on the formula .

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

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

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

The greatest of the angle measures is given by .

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

Compare your answer with the correct one above

Question

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

Answer

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

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

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

Compare your answer with the correct one above

Question

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

Answer

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

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

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

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

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Question

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

Answer

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

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

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

Compare your answer with the correct one above

Question

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

Answer

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

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

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

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

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Question

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

Answer

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

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

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

Compare your answer with the correct one above

Question

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

Answer

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

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

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

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

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