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Amy Ferguson Moncure

H Algebra 2/Trig · Unit 8

Arithmetic Terms

Use the common difference to jump from a known term to the term you need.

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Key Ideas Easy Medium Hard

an = a1 + (n − 1)d

an = ak + (n − k)d

What to Remember

From the first term

If an = an−1 + d and you know a1, then

an = a1 + (n − 1)d.

From a different term

If you know ak and d, step forward or backward.

an = ak + (n − k)d.

Level 1

Easy

The first term and the common difference

Example
A sequence is defined by an = an−1 + 5, with a1 = −8. Find a5.

an = a1 + (n − 1)d

a5 = −8 + (5 − 1)(5)

a5 = −8 + 20

a5 = 12

Answer: a5 = 12

1
A sequence is defined by an = an−1 + 3, with a1 = −4. Find a6.
Show Answer

an = a1 + (n − 1)d

a6 = −4 + (6 − 1)(3)

a6 = −4 + 15

a6 = 11

Answer: a6 = 11

2
A sequence is defined by an = an−1 + 6, with a1 = 2. Find a4.
Show Answer

an = a1 + (n − 1)d

a4 = 2 + (4 − 1)(6)

a4 = 2 + 18

a4 = 20

Answer: a4 = 20

3
A sequence is defined by an = an−1 − 2, with a1 = 15. Find a7.
Show Answer

an = a1 + (n − 1)d

a7 = 15 + (7 − 1)(−2)

a7 = 15 − 12

a7 = 3

Answer: a7 = 3

4
A sequence is defined by an = an−1 + 4, with a1 = −11. Find a5.
Show Answer

an = a1 + (n − 1)d

a5 = −11 + (5 − 1)(4)

a5 = −11 + 16

a5 = 5

Answer: a5 = 5

5
A sequence is defined by an = an−1 − 5, with a1 = 9. Find a6.
Show Answer

an = a1 + (n − 1)d

a6 = 9 + (6 − 1)(−5)

a6 = 9 − 25

a6 = −16

Answer: a6 = −16

Level 2

Medium

One later term and the common difference

Example
An arithmetic sequence has a6 = 4 and d = 3. Find a10.

an = ak + (n − k)d

a10 = 4 + (10 − 6)(3)

a10 = 4 + 12

a10 = 16

Answer: a10 = 16

1
An arithmetic sequence has a4 = 18 and d = −2. Find a9.
Show Answer

an = ak + (n − k)d

a9 = 18 + (9 − 4)(−2)

a9 = 18 − 10

a9 = 8

Answer: a9 = 8

2
An arithmetic sequence has a8 = −7 and d = 4. Find a3.
Show Answer

an = ak + (n − k)d

a3 = −7 + (3 − 8)(4)

a3 = −7 − 20

a3 = −27

Answer: a3 = −27

3
An arithmetic sequence has a5 = 12 and d = 7. Find a8.
Show Answer

an = ak + (n − k)d

a8 = 12 + (8 − 5)(7)

a8 = 12 + 21

a8 = 33

Answer: a8 = 33

4
An arithmetic sequence has a3 = −6 and d = −4. Find a10.
Show Answer

an = ak + (n − k)d

a10 = −6 + (10 − 3)(−4)

a10 = −6 − 28

a10 = −34

Answer: a10 = −34

5
An arithmetic sequence has a7 = 20 and d = 5. Find a2.
Show Answer

an = ak + (n − k)d

a2 = 20 + (2 − 7)(5)

a2 = 20 − 25

a2 = −5

Answer: a2 = −5

Level 3

Hard

The first term and a much later term

Example
An arithmetic sequence has a5 = −9 and d = 4. Find a1 and a12.

an = ak + (n − k)d

Find the first term.

a1 = −9 + (1 − 5)(4)

a1 = −9 − 16

a1 = −25

Use the same formula for the later term.

a12 = −9 + (12 − 5)(4)

a12 = −9 + 28

a12 = 19

Answer: a1 = −25 and a12 = 19

1
An arithmetic sequence has a6 = 11 and d = −3. Find a1 and a15.
Show Answer

an = ak + (n − k)d

Find the first term.

a1 = 11 + (1 − 6)(−3)

a1 = 11 + 15

a1 = 26

Use the same formula for the later term.

a15 = 11 + (15 − 6)(−3)

a15 = 11 − 27

a15 = −16

Answer: a1 = 26 and a15 = −16

2
An arithmetic sequence has a4 = 2 and d = 6. Find a1 and a11.
Show Answer

an = ak + (n − k)d

Find the first term.

a1 = 2 + (1 − 4)(6)

a1 = 2 − 18

a1 = −16

Use the same formula for the later term.

a11 = 2 + (11 − 4)(6)

a11 = 2 + 42

a11 = 44

Answer: a1 = −16 and a11 = 44

3
An arithmetic sequence has a8 = −4 and d = 5. Find a1 and a14.
Show Answer

an = ak + (n − k)d

Find the first term.

a1 = −4 + (1 − 8)(5)

a1 = −4 − 35

a1 = −39

Use the same formula for the later term.

a14 = −4 + (14 − 8)(5)

a14 = −4 + 30

a14 = 26

Answer: a1 = −39 and a14 = 26

4
An arithmetic sequence has a3 = 21 and d = −4. Find a1 and a10.
Show Answer

an = ak + (n − k)d

Find the first term.

a1 = 21 + (1 − 3)(−4)

a1 = 21 + 8

a1 = 29

Use the same formula for the later term.

a10 = 21 + (10 − 3)(−4)

a10 = 21 − 28

a10 = −7

Answer: a1 = 29 and a10 = −7

5
An arithmetic sequence has a7 = −15 and d = 3. Find a1 and a16.
Show Answer

an = ak + (n − k)d

Find the first term.

a1 = −15 + (1 − 7)(3)

a1 = −15 − 18

a1 = −33

Use the same formula for the later term.

a16 = −15 + (16 − 7)(3)

a16 = −15 + 27

a16 = 12

Answer: a1 = −33 and a16 = 12

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