Series Binomial Theorem Limits Test Answer Key with Scoring – Precalculus

Question 1 Evaluating a Sum (2 points)

The terms form an arithmetic sequence: a₁ = 2, a₈ = 37. Sum = 82(2 + 37) = 4(39) = 156.

Answer: 156

Scoring: 1 point for identifying the first and last terms, 1 point for the sum.

Question 2 Twenty-five Terms (2 points)

a₂₅ = −6 + 24(4) = 90. S₂₅ = 252(−6 + 90) = 252(84) = 1,050.

Answer: 1,050

Scoring: 1 point for the 25th term, 1 point for the sum.

Question 3 A Geometric Sum (2 points)

S₈ = 6 · 1 − (−2)⁸1 − (−2) = 6 · 1 − 2563 = 6 · (−85) = −510.

Answer: −510

Scoring: 1 point for the formula with correct values, 1 point for the result.

Question 4 Infinite Series (3 points)

(a) r = −12 ÷ 18 = −23. Since |r| < 1, the series converges.

(b) S = 181 + 23 = 1853 = 545 = 10.8.

(c) 0.181818… = 0.18 + 0.0018 + …, with a₁ = 0.18 and r = 0.01, so S = 0.180.99 = 211.

Answer: (a) r = −23, converges; (b) 545; (c) 211

Scoring: 1 point for each part.

Question 5 Expanding a Binomial (2 points)

The coefficients are 1, 4, 6, 4, 1. (x − 2)⁴ = x⁴ + 4(−2)x³ + 6(4)x² + 4(−8)x + 16.

Answer: x⁴ − 8x³ + 24x² − 32x + 16

Scoring: 1 point for correct coefficients and powers, 1 point for correct signs and constants.

Question 6 A Term of an Expansion (2 points)

The fourth term has k = 3: C(5, 3)(3x)²(−y)³ = 10 · 9x² · (−y³).

Answer: −90x²y³

Scoring: 1 point for the setup with C(5, 3), 1 point for the term.

Question 7 Reading a Graph (3 points)

(a) As x approaches 3 from either side, g(x) approaches 3 + 1 = 4.

(b) The filled dot is at (3, 1), so g(3) = 1.

(c) No. The limit (4) does not equal the function value (1), so the graph has a jump at x = 3.

Answer: (a) 4; (b) 1; (c) not continuous

Scoring: 1 point for each part.

Question 8 Two Limits (2 points)

(a) x² + x − 2 = (x + 2)(x − 1), so for x ≠ −2 the expression is x − 1, which approaches −3.

(b) Divide the top and bottom by n: 4 + 7/n1 + 1/n → 41 = 4.

Answer: (a) −3; (b) 4

Scoring: 1 point for each part.

Question 9 Bouncing Ball (2 points)

The first drop is 8 ft. The rebound heights 6, 4.5, … form a geometric series with a₁ = 6 and r = 0.75, and each rebound is traveled twice.

Up-and-down distance = 2 · 61 − 0.75 = 2(24) = 48. Total = 8 + 48 = 56 ft.

Answer: 56 feet

Scoring: 1 point for the series setup, 1 point for the total.