Number Concepts and Operations - SSAT Upper Level Quantitative

Card 0 of 2248

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

Find the next term of the arithmetic sequence:

Answer

The common difference for this sequence is . To find the next number in the sequence, add to the last given number.

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Question

Find the next term in the following arithmetic sequence:

Answer

The sequence is subtracting for each term. The next term can be found by subtracting from :

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Question

Find the next term in the following arithmetic sequence:

$\frac{9}{2}, 4, \frac{7}{2}, ?$

Answer

The common difference for this arithmetic sequence is $-\frac{1}{2}$ . To find the next term in the sequence, subtract that value from the last given value.

$\frac{7}{2}-\frac{1}{2}=3$

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Question

Find the next number in this arithmetic sequence:

Answer

The common difference for this arithmetic sequence is . Take away from to get the next term.

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Question

Find the next term in this arithmetic sequence:

Answer

The common difference for this sequence is . Subtract from the last given term to find the next term.

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Question

Find the next term of this arithmetic sequence:

Answer

The common difference for this sequence is . Add this to the last given term to find the next one.

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Question

Find the next term of the arithmetic sequence:

Answer

The common difference is . Add this to the last given term to find the next term.

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Question

Find the next term of the arithmetic sequence:

$-\frac{1}{2}, 3, \frac{13}{2}, ?$

Answer

The common difference is $\frac{7}{2}$ . Add this to the last given term to find the next term.

$\frac{13}{2}+\frac{7}{2}=\frac{20}{2}=10$

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Question

Find the next term of the arithmetic sequence:

Answer

The common difference is . Add this to the last given term to find the next term.

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Question

Find the next term of the arithmetic sequence:

Answer

The common difference is . Add this to the last given term to find the next term.

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Question

Find the next term of the arithmetic sequence:

Answer

The common difference is . Add this to the last given term to find the next term.

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Question

The first two terms of an arithmetic sequence are 4 and 9, in that order. Give the one-hundredth term of that sequence.

Answer

The first term is $x_{0} = 4$ ; the common difference is

.

The hundredth term is

$x_{99} = x_{0} + 99 d = 4 + 99 \cdot5 = 4 + 495 = 499$ .

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Question

The first two terms of an arithmetic sequence are 1,000 and 997, in that order. What is the seventieth term?

Answer

The first term is $x_{0} = 1,000$ .

The common difference is

.

The seventieth term is

$x_{69} = x_{0}+ 69 \cdot d= 1,000 + 69 \cdot (-3)= 1,000 - 207 = 793$ .

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Question

An arithmetic sequence begins as follows:

Which of the following terms is the first positive term in the sequence?

Answer

The common difference of the sequence is

,

so the th term of the sequence is

$a_{n} = -128 + 3.3 (n-1)$

To find out the minimum value for which $a_{n} > 0$ , set up this inequality:

The first positive term is the fortieth term.

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Question

An arithmetic sequence begins as follows:

$-78 \frac{2}{3}, -75 \frac{4}{5},...$

Which of the following terms is the first positive term in the sequence?

Answer

The common difference of the sequence is

$-75 \frac{4}{5} - \left ( -78 \frac{2}{3} \right ) =2 \frac{13}{15}$ ,

so the th term of the sequence is

$a_{n} = - 78\frac{2}{3} + \left ( 2\frac{13}{15} \right ) (n-1)$

To find out the minimum value for which $a_{n} > 0$ , set up this inequality:

$- 78\frac{2}{3} + \left ( 2\frac{13}{15} \right ) (n-1) > 0$

$- 78\frac{2}{3} + \left ( 2\frac{13}{15} \right ) n - 2\frac{13}{15} > 0$

$\left ( 2\frac{13}{15} \right ) n - 81\frac{8}{15} > 0$

$\left ( 2\frac{13}{15} \right ) n > 81\frac{8}{15}$

$n > 81\frac{8}{15} \div 2\frac{13}{15} = 28 \frac{19}{43}$

The first positive term in the sequence is the twenty-ninth term.

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Question

An arithmetic sequence begins as follows:

$155 \frac{5}{6}, 153 \frac{3}{4},...$

Which of the following terms is the first negative term in the sequence?

Answer

The common difference of the sequence is

$153 \frac{3}{4}- 155 \frac{5}{6}= -2 \frac{1}{12}$

so the th term of the sequence is

$a_{n} = 155\frac{5}{6}-\left (2 \frac{1}{12} \right )(n-1)$

To find out the minimum value for which $a_{n} < 0$ , set up this inequality:

$155\frac{5}{6}-\left (2 \frac{1}{12} \right )(n-1) < 0$

$155\frac{5}{6}-\left (2 \frac{1}{12} \right ) n + 2\frac{1}{12} < 0$

$157\frac{11}{12}-\left (2 \frac{1}{12} \right ) n < 0$

$157\frac{11}{12} < \left (2 \frac{1}{12} \right ) n$

$\left (2 \frac{1}{12} \right ) n > 157\frac{11}{12}$

$n > 157\frac{11}{12} \div 2 \frac{1}{12} = 75 \frac{4}{5}$

The seventy-sixth term is the first negative term.

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