Sequences and Series: Algebra 2 Math Test Answer Key – PDF Download

Question 1: Sequence from a graph (2 points)

1. The points rise 4 each step: \(a_{n}=5+4(n-1)=\)\(4n+1\).
2. \(a_{30}=4(30)+1=\)\(121\).

Scoring: 1 point for the formula (accept \(a_{n}=5+4(n-1)\)); 1 point for 121.

Question 2: Alternating sequence (2 points)

1. Ratio \(\frac{-12}{6}=-2\): \(a_{n}=6(-2)^{n-1}\).
2. \(a_{9}=6(-2)^{8}=6\cdot 256=\)\(1536\).

Scoring: 1 point for the formula with the correct ratio; 1 point for 1536 (the sign is positive because the exponent 8 is even).

Question 3: Recursive to explicit (2 points)

1. \(9,\,5,\,1,\,-3\).
2. \(a_{n}=9-4(n-1)=13-4n\), so \(a_{25}=13-100=\)\(-87\).

Scoring: 1 point for four correct terms; 1 point for a correct explicit formula and \(a_{25}=-87\).

Question 4: Sigma evaluation (2 points)

1. \(5+7+9+11+13+15\).
2. \(5+7+9+11+13+15=\)\(60\). (Check: \(\frac{6(5+15)}{2}=60\).)

Scoring: 1 point for the six terms; 1 point for the sum 60.

Question 5: Lecture hall (2 points)

1. \(a_{20}=12+19(3)=\)\(69\) seats.
2. \(S_{20}=\frac{20(12+69)}{2}=10\cdot 81=\)\(810\) seats.

Scoring: 1 point for 69; 1 point for 810 (award it if the method is right and only the arithmetic from part a carries an error).

Question 6: Series with negative ratio (2 points)

Geometric with \(a_{1}=3\), \(r=-2\), \(n=8\): \(S_{8}=\frac{3(1-(-2)^{8})}{1-(-2)}=\frac{3(1-256)}{3}=\)\(-255\).

Scoring: 1 point for identifying \(a_{1}=3\), \(r=-2\), \(n=8\) and substituting; 1 point for \(-255\).

Question 7: Missing terms (2 points)

1. \(r^{2}=\frac{96}{24}=4\), so \(r=2\) (positive terms). \(a_{1}=\frac{24}{2^{3}}=\)\(3\).
2. \(a_{8}=3\cdot 2^{7}=\)\(384\).

Scoring: 1 point for \(r=2\) and \(a_{1}=3\); 1 point for 384.

Question 8: Growing deposits (3 points)

1. \(a_{24}=100+23(15)=\)$445.
2. \(S_{24}=\frac{24(100+445)}{2}=12\cdot 545=\)$6,540.

Scoring: 1 point for $445; 1 point for a correct series formula with the right values; 1 point for $6,540.

Question 9: Loan amount (3 points)

1. \(\sum_{k=1}^{24}400(1.005)^{-k}\).
2. Geometric with \(a_{1}=\frac{400}{1.005}\), \(r=\frac{1}{1.005}\): \(P=400\cdot\frac{1-1.005^{-24}}{0.005}\approx \$9,025.15\), so $9,025.15.
3. \(400\cdot 24=9600\) paid, and \(9600-9025.15\approx \$574.85\). $574.85 of interest.

Scoring: 1 point for the sigma expression; 1 point for the amount borrowed; 1 point for the interest.

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