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SSAT Upper Level Math : How to find the nth term of an arithmetic sequence

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Example Questions

Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

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

Possible Answers:

796

790

793

1,207

1,210

Correct answer:

793

Explanation:

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

The common difference is

997 - 1,000 = -3 .

The seventieth term is

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

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Example Question #2 : How To Find The Nth Term Of An Arithmetic Sequence

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

Possible Answers:

504

899

904

499

494

Correct answer:

499

Explanation:

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

d = 9-4 = 5 .

The hundredth term is

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

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Example Question #3 : How To Find The Nth Term Of An Arithmetic Sequence

An arithmetic sequence begins as follows:

-128, -124.7,...

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

Possible Answers:

The thirty-seventh term

The sequence has no positive terms.

The thirty-ninth term

The thirty-eighth term

The fortieth term

Correct answer:

The fortieth term

Explanation:

The common difference of the sequence is

-124.7 - (-128) = 3.3 ,

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:

-128 + 3.3 (n-1) > 0

-128 + 3.3 n-3.3 > 0

3.3 n-131.3 > 0

3.3 n>131.3

n > 39.8

The first positive term is the fortieth term.

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Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

An arithmetic sequence begins as follows:

5.7, 8.1,...

Which of the following is the first term greater than 100?

Possible Answers:

The forty-fourth term

The forty-first term

The forty-second term

The forty-third term

The fortieth term

Correct answer:

The forty-first term

Explanation:

The common difference of the sequence is

8.1 - 5.7 = 2.4

so the th term of the sequence is

$a_{n} =5.7+2.4 (n-1)$

To find out the minimum value for which $a_{n} > 100$ , set up this inequality:

5.7+2.4 (n-1) > 100

5.7+2.4 n-2.4 > 100

2.4 n +3.3 > 100

2.4 n > 96.7

$n > 96.7 \div 2.4 \approx 40.3$

The forty-first term is the correct response.

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Example Question #3 : Nth Term Of An Arithmetic Sequence

An arithmetic sequence begins as follows:

300, 297.3,...

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

Possible Answers:

The one hundred thirteenth term

The one hundred twelfth term

The one hundred tenth term

The one hundred eleventh term

The one hundred fourteenth term

Correct answer:

The one hundred thirteenth term

Explanation:

The common difference of the sequence is

297.3 - 300 = -2.7

so the th term of the sequence is

$a_{n} = 300 -2.7 (n-1)$

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

300 -2.7 (n-1) < 0

300 -2.7 n+2.7 < 0

302.7-2.7 n < 0

302.7 < 2.7 n

2.7 n > 302.7

$n > 302.7 \div 2.7 \approx 112.1$

The first negative term is the one hundred thirteenth term.

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Example Question #6 : How To Find The Nth Term Of An Arithmetic Sequence

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?

Possible Answers:

The seventy-fourth term

The seventy-sixth term

The seventy-seventh term

The seventy-fifth term

The seventy-eighth term

Correct answer:

The seventy-sixth term

Explanation:

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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Example Question #4 : How To Find The Nth Term Of An Arithmetic Sequence

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?

Possible Answers:

The thirtieth term

The twenty-eighth term

The twenty-ninth term

The twenty-seventh term

The sequence has no positive terms.

Correct answer:

The twenty-ninth term

Explanation:

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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Example Question #7 : Nth Term Of An Arithmetic Sequence

An arithmetic sequence begins as follows:

$1 \frac{1}{7}, 3 \frac{1}{4},...$

Which of the following is the first term greater than 100?

Possible Answers:

The forty-eighth term

The forty-ninth term

The fifty-first term

The forty-seventh term

The fiftieth term

Correct answer:

The forty-eighth term

Explanation:

The common difference of the sequence is

$3 \frac{1}{4} - 1 \frac{1}{7} = 2\frac{3}{28}$

so the th term of the sequence is

$a_{n} = 1\frac{1}{7} + \left ( 2\frac{3}{28} \right ) (n-1)$

To find out the minimum value for which $a_{n} > 100$ , set up this inequality:

$1\frac{1}{7} + \left ( 2\frac{3}{28} \right ) (n-1) > 100$

$1\frac{1}{7} + \left ( 2\frac{3}{28} \right ) n -2\frac{3}{28} > 100$

$\left ( 2\frac{3}{28} \right ) n - \frac{27}{28} > 100$

$\left ( 2\frac{3}{28} \right ) n > 100 \frac{27}{28}$

$n > 100 \frac{27}{28} \div 2\frac{3}{28} = 47 \frac{54}{59}$

The correct response is the forty-eighth term.

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Example Question #3 : How To Find The Nth Term Of An Arithmetic Sequence

The tenth and twelfth terms of an arithmetic sequence are 8.4 and 10.2. What is its first term?

Possible Answers:

0.9

1.2

1.8

0.3

-0.6

Correct answer:

0.3

Explanation:

The th term of an arithmetic sequence with initial term $a_{1}$ and common difference is defined by the equation

$a_{n} = a_{1}+ (n-1)d$

Since the tenth and twelfth terms are two terms apart, the common difference can be found as follows:

$a_{10}+2d=a_{12}$

8.4+2d=10.2

8.4+2d- 8.4=10.2 - 8.4

2d=1.8

d = 0.9

Now, we can set n = 10, d = 0.9 in the sequence equation to find $a_{1}$ :

$a_{n} = a_{1}+ (n-1)d$

$a_{10} = a_{1}+ (10-1)0.9$

$8.4 = a_{1}+ 9 \cdot 0.9$

$8.4 = a_{1}+ 8.1$

$8.4 - 8.1 = a_{1}+ 8.1 - 8.1$

$a_{1} = 0.3$

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Example Question #10 : How To Find The Nth Term Of An Arithmetic Sequence

The eleventh and thirteenth terms of an arithmetic sequence are, respectively, 11 and 14. Give its first term.

Possible Answers:

$-5\frac{1}{2}$

$-2\frac{1}{2}$

$1\frac{1}{2}$

Correct answer:

Explanation:

The th term of an arithmetic sequence with initial term $a_{1}$ and common difference is defined by the equation

$a_{n} = a_{1}+ (n-1)d$

Since the eleventh and thirteenth terms are two terms apart, the common difference can be found as follows:

$a_{11}+2d=a_{13}$

11+2d=14

11+2d -11=14 -11

2d = 3

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

Now, we can set $n = 11, d =\frac{3}{2}$ in the sequence equation to find $a_{1}$ :

$a_{11} = a_{1}+ (11-1) \frac{3}{2}$

$11= a_{1}+ 10 \cdot \frac{3}{2}$

$11= a_{1}+ 15$

$11- 15= a_{1}+ 15 - 15$

$a_{1} = -4$

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SSAT Upper Level Math : How to find the nth term of an arithmetic sequence

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

All SSAT Upper Level Math Resources

Example Questions

Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

Example Question #2 : How To Find The Nth Term Of An Arithmetic Sequence

Example Question #3 : How To Find The Nth Term Of An Arithmetic Sequence

Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

Example Question #3 : Nth Term Of An Arithmetic Sequence

Example Question #6 : How To Find The Nth Term Of An Arithmetic Sequence

Example Question #4 : How To Find The Nth Term Of An Arithmetic Sequence

Example Question #7 : Nth Term Of An Arithmetic Sequence

Example Question #3 : How To Find The Nth Term Of An Arithmetic Sequence

Example Question #10 : How To Find The Nth Term Of An Arithmetic Sequence

All SSAT Upper Level Math Resources

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