Arithmetic and Geometric Sequences: Algebra 1 Math Worksheet Answer Key – PDF Download

Algebra 1 Sequences Worksheet Answer Key: problems 1–5

Problem 1: Arithmetic: next terms

\(9-5=4\), \(13-9=4\), \(17-13=4\). The common difference is \(d=4\). The next terms are \(21\) and \(25\).

Problem 2: Geometric: next terms

\(12: 4=3\), \(36: 12=3\), \(108: 36=3\). The common ratio is \(r=3\). Next: \(108\cdot 3=324\) and \(324\cdot 3=972\). The terms are \(324\) and \(972\).

Problem 3: Decreasing arithmetic

\(43-50=-7\) and \(36-43=-7\), so \(d=-7\). The next terms are \(36-7=29\), \(29-7=22\) and \(22-7=15\): \(29, 22, 15\).

Problem 4: A ratio less than 1

\(40: 80=\frac{1}{2}\) and \(20: 40=\frac{1}{2}\), so \(r=\frac{1}{2}\). Halve each term: \(10\), \(5\) and \(2.5\). The next terms are \(10, 5, 2.5\).

Problem 5: Arithmetic, geometric or neither

1. Differences: \(2,4,6\) (not equal). Ratios: \(2,2,1.75\) (not equal). Neither.
2. Ratios: \(\frac{-2}{1}=-2\), \(\frac{4}{-2}=-2\), \(\frac{-8}{4}=-2\). Geometric with \(r=-2\).
3. Differences: \(-5,-5,-5\). Arithmetic with \(d=-5\).

Algebra 1 Sequences Worksheet Answer Key: problems 6–10

Problem 6: Explicit rule: arithmetic

\(a_{1}=6\) and \(d=5\). Then \(a_{n}=6+5(n-1)=5n+1\). The formula is \(a_{n}=6+5(n-1)\).

\(a_{20}=6+5(19)=6+95=101\). The 20th term is \(101\).

Problem 7: Explicit rule: geometric

\(a_{1}=5\) and \(r=2\). The formula is \(a_{n}=5\cdot 2^{n-1}\).

\(a_{8}=5\cdot 2^{7}=5\cdot 128=640\). The 8th term is \(640\).

Problem 8: Recursive: arithmetic

\(d=17-20=-3\). The recursive formula is \(a_{1}=20,\quad a_{n}=a_{n-1}-3\).

Continue: \(11-3=8\), \(8-3=5\). So \(a_{6}=5\). Check: \(20+5(-3)=5\).

Problem 9: Recursive: geometric

\(r=3: 1.5=2\). The recursive formula is \(a_{1}=1.5,\quad a_{n}=2\cdot a_{n-1}\).

The terms are \(1.5,3,6,12,24,48,96\), so \(a_{7}=96\). Check: \(1.5\cdot 2^{6}=1.5\cdot 64=96\).

Problem 10: Use a_1 and d

\(a_{n}=-8+6(n-1)=6n-14\). The formula is \(a_{n}=-8+6(n-1)\).

\(a_{15}=-8+6(14)=-8+84=76\). So \(a_{15}=76\).

Algebra 1 Sequences Worksheet Answer Key: problems 11–15

Problem 11: From a recursive rule

1. \(a_{1}=4\), \(a_{2}=12\), \(a_{3}=36\), \(a_{4}=108\), \(a_{5}=324\).
2. The sequence is geometric with \(r=3\): \(a_{n}=4\cdot 3^{n-1}\). Then \(a_{5}=4\cdot 3^{4}=4\cdot 81=324\), which matches.

Problem 12: Find the rule from two terms

From term 3 to term 7 there are 4 steps, so \(4d=27-11=16\) and \(d=4\). Then \(a_{1}=a_{3}-2d=11-8=3\). The formula is \(a_{n}=3+4(n-1)=4n-1\).

Check: \(a_{3}=4(3)-1=11\) and \(a_{7}=4(7)-1=27\).

Problem 13: Geometric from two terms

From term 2 to term 5 the ratio is used 3 times : \(r^{3}=\frac{162}{6}=27\), so \(r=3\). Then \(a_{1}=6: 3=2\). The formula is \(a_{n}=2\cdot 3^{n-1}\).

Check: \(a_{5}=2\cdot 3^{4}=2\cdot 81=162\).

Problem 14: Which term?

\(a_{n}=4+6(n-1)\). Set it equal to 154: \(4+6(n-1)=154\), so \(6(n-1)=150\) and \(n-1=25\). Then \(n=26\). The 26th term is 154.

Problem 15: Sequence from a graph

1. The terms are \(10,7,4,1\). The differences are all \(-3\), so it is arithmetic with \(d=-3\).
2. \(a_{n}=10-3(n-1)=13-3n\). Then \(a_{8}=13-3(8)=-11\), so \(a_{8}=-11\).

Algebra 1 Sequences Worksheet Answer Key: problems 16–20

Problem 16: Decay by a third

\(a_{n}=243(\frac{1}{3})^{n-1}\). So \(a_{6}=243(\frac{1}{3})^{5}=\frac{243}{243}=1\). The answer is \(a_{6}=1\). Check by listing: \(243,81,27,9,3,1\).

Problem 17: Weekly savings

1. \(a_{n}=15+6(n-1)=6n+9\).
2. \(a_{12}=15+6(11)=15+66=81\). He deposits $81.
3. \(15+6(n-1)>100\) gives \(6(n-1)>85\), so \(n-1>14.17\) and \(n=16\). Check: \(a_{15}=99\) is too small and \(a_{16}=105\) works. It is week 16.

Problem 18: Doubling views

1. Recursive: \(a_{1}=50,\quad a_{n}=2\cdot a_{n-1}\). Explicit: \(a_{n}=50\cdot 2^{n-1}\).
2. \(a_{7}=50\cdot 2^{6}=50\cdot 64=3200\). The video gets 3,200 views on day 7.

Problem 19: Stadium rows

1. \(a_{n}=18+2(n-1)=2n+16\).
2. \(a_{25}=18+2(24)=66\). Row 25 has 66 seats.
3. \(18+2(n-1)=50\), so \(2(n-1)=32\) and \(n=17\). It is row 17.

Problem 20: Car depreciation

1. Losing 15% leaves 85%, so \(r=0.85\): \(a_{n}=20000(0.85)^{n-1}\).
2. \(a_{4}=20000(0.85)^{3}=20000(0.614125)=12282.5\). The value is $12,282.50.
3. \(a_{5}=20000(0.85)^{4}\approx 10440.13\) is still above $10,000, and \(a_{6}=20000(0.85)^{5}\approx 8874.11\) is below. It happens at the start of year 6.

Algebra 1 Sequences Worksheet Answer Key: problems 21–22

Problem 21: Two growth plans

1. Plan A (arithmetic): \(A_{n}=100+25(n-1)\). Plan B (geometric): \(B_{n}=10\cdot 2^{n-1}\).
2. Compare month by month. Month 5: \(A_{5}=200\) and \(B_{5}=160\). Month 6: \(A_{6}=225\) and \(B_{6}=320\). Plan B first exceeds Plan A in month 6.

Problem 22: Bouncing ball

1. \(a_{n}=16(\frac{3}{4})^{n-1}\).
2. \(a_{4}=16(\frac{3}{4})^{3}=16\cdot\frac{27}{64}=\frac{27}{4}=6.75\). The 4th bounce reaches 6.75 feet.

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