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

Question 1: Identify the sequence (2 points)

1. Differences: \(7,7,7\). Arithmetic, \(d=7\), next term \(33+7=40\).
2. Ratios: \(-2,-2,-2\). Geometric, \(r=-2\), next term \(-24\cdot(-2)=48\).

Scoring: 1 point for each part: the correct classification with its difference or ratio, and the correct next term.

Question 2: Arithmetic formula (2 points)

1. \(d=-4\), so \(a_{n}=9-4(n-1)\) (equivalently \(a_{n}=13-4n\)).
2. \(a_{12}=9-4(11)=9-44=-35\).

Scoring: 1 point for a correct formula (any equivalent form); 1 point for \(a_{12}=-35\).

Question 3: Geometric formula (2 points)

1. \(r=14: 7=2\), so \(a_{n}=7\cdot 2^{n-1}\).
2. \(a_{9}=7\cdot 2^{8}=7\cdot 256=1792\).

Scoring: 1 point for the formula with the correct ratio and exponent \(n-1\); 1 point for the value 1792.

Question 4: Recursive formulas (2 points)

1. \(d=85-100=-15\): \(a_{1}=100,\quad a_{n}=a_{n-1}-15\).
2. \(a_{2}=10\), \(a_{3}=50\), \(a_{4}=250\).

Scoring: 1 point for part (a), including the starting value and the rule; 1 point for \(a_{4}=250\).

Question 5: Sequence from a table (3 points)

1. \(23-14=9\), \(32-23=9\), \(41-32=9\). So \(d=9\).
2. \(a_{n}=14+9(n-1)=9n+5\). The formula is \(a_{n}=14+9(n-1)\).
3. \(9n+5=140\), so \(9n=135\) and \(n=15\). The 15th term is 140.

Scoring: 1 point for each part; in part (c) the equation must be set up and solved for \(n\).

Question 6: Sequence from a graph (3 points)

1. The terms are \(5,10,20,40\), so \(r=10: 5=2\).
2. \(a_{n}=5\cdot 2^{n-1}\).
3. \(a_{7}=5\cdot 2^{6}=5\cdot 64=320\).

Scoring: 1 point for each part.

Question 7: Arithmetic from two terms (2 points)

1. There are 5 steps from term 4 to term 9: \(5d=44-19=25\), so \(d=5\).
2. \(a_{1}=19-3(5)=4\), so \(a_{n}=4+5(n-1)=5n-1\). Check: \(a_{9}=5(9)-1=44\).

Scoring: 1 point for the common difference; 1 point for the formula (any equivalent form).

Question 8: Geometric from two terms (2 points)

1. From term 2 to term 4 the ratio is used twice: \(r^{2}=\frac{48}{12}=4\), and \(r>0\), so \(r=2\).
2. \(a_{5}=96\) and \(a_{6}=192\). Also \(a_{1}=6\) and \(6\cdot 2^{5}=192\). So \(a_{6}=192\).

Scoring: 1 point for \(r=2\) (with the positive root chosen); 1 point for \(a_{6}=192\).

Question 9: Growing plant (2 points)

1. \(a_{n}=4+1.5(n-1)\). \(a_{9}=4+1.5(8)=16\) inches.
2. \(4+1.5(n-1)>30\) gives \(1.5(n-1)>26\), so \(n-1>17.33\) and \(n=19\). Check: week 18 is 29.5 inches, week 19 is 31 inches. It is week 19.

Scoring: 1 point for the formula and week-9 height; 1 point for week 19 with a correct check or inequality.

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