How to find the nth term of an arithmetic sequence - SSAT Upper Level Quantitative

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An arithmetic sequence begins as follows:

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

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The common difference of the sequence is

so the th term of the sequence is

To find out the minimum value for which , set up this inequality:

$n > 302.7 \div 2.7 \approx 112.1$

The first negative term is the one hundred thirteenth term.

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?

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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

To find out the minimum value for which , 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.

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

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The first term is ; the common difference is

.

The hundredth term is

.

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

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The first term is .

The common difference is

.

The seventieth term is

.

An arithmetic sequence begins as follows:

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

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The common difference of the sequence is

,

so the th term of the sequence is

To find out the minimum value for which , set up this inequality:

The first positive term is the fortieth term.

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?

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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

To find out the minimum value for which , 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.

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?

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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

To find out the minimum value for which , 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.

An arithmetic sequence begins as follows:

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

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The common difference of the sequence is

so the th term of the sequence is

To find out the minimum value for which , set up this inequality:

$n > 96.7 \div 2.4 \approx 40.3$

The forty-first term is the correct response.

An arithmetic sequence begins as follows:

Give the thirty-second term of this sequence.

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The th term of an arithmetic sequence with initial term and common difference is defined by the equation

The initial term in the given sequence is

;

the common difference is

;

We are seeking term .

This term is

An arithmetic sequence begins as follows:

$$\frac{2}{5}$ , $\frac{7}{10}$, 1, 1 $\frac{3}{10}$, ...$

Give the thirty-third term of this sequence.

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The th term of an arithmetic sequence with initial term and common difference is defined by the equation

.

The initial term in the given sequence is

;

the common difference is

$d = $\frac{7}{10}$ - $\frac{2}{5}$ = $\frac{7}{10}$ - $\frac{4}{10}$ = $\frac{3}{10}$$ .

We are seeking term .

Therefore,

$= $\frac{2}{5}$+ (32) $\frac{3}{10}$$

$= $\frac{2}{5}$+ $\frac{96}{10}$$

$= $\frac{2}{5}$+ $\frac{48}{5}$$

$= $\frac{50}{5}$ = 10$ ,

which is not among the choices.

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

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The th term of an arithmetic sequence with initial term and common difference is defined by the equation

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

Now, we can set in the sequence equation to find :

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

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

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

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

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The th term of an arithmetic sequence with initial term and common difference is defined by the equation

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

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

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

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

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

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

The lengths of the sides of ten squares form an arithmetic sequence. One side of the smallest square measures eight inches; one side of the second-smallest square measures one foot.

Give the area of the largest square.

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Let be the lengths of the sides of the squares in inches. and , so their common difference is

$d = $a_{2}$ - $a_{1}$ = 12 - 8 = 4$

The arithmetic sequence formula is

The length of a side of the largest square - square 10 - can be found by substituting :

The largest square has sides of length 44 inches, so its area is the square of this, or $A = $44^{2}$ = 1,936$ square inches.

An arithmetic sequence begins as follows:

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

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The common difference of the sequence is

so the th term of the sequence is

To find out the minimum value for which , set up this inequality:

$n > 302.7 \div 2.7 \approx 112.1$

The first negative term is the one hundred thirteenth term.

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?

Tap to see back →

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

To find out the minimum value for which , 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.

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

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The first term is ; the common difference is

.

The hundredth term is

.

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

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The first term is .

The common difference is

.

The seventieth term is

.

An arithmetic sequence begins as follows:

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

Tap to see back →

The common difference of the sequence is

,

so the th term of the sequence is

To find out the minimum value for which , set up this inequality:

The first positive term is the fortieth term.

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?

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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

To find out the minimum value for which , 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.

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?

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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

To find out the minimum value for which , 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.