What is the common difference in the arithmetic sequence 5, 11, 17, 23, …? — 6.Common difference d = 11 − 5 = 6. Each term increases by 6.
What is the 10th term of the arithmetic sequence with a₁ = 3 and d = 4? — 39.aₙ = a₁ + (n−1)d = 3 + 9·4 = 3 + 36 = 39.
Which of the following is a geometric sequence? — 3, 9, 27, 81.In 3, 9, 27, 81 each term is multiplied by 3, giving a constant ratio r = 3.
What is the sum of the first 5 terms of the arithmetic series with a₁ = 2 and d = 3? — 35.Sₙ = n/2·(2a₁ + (n−1)d) = 5/2·(4 + 12) = 5/2·16 = 40. Wait — let me recalculate: a₅ = 2+4·3=14, S₅ = 5/2·(2+14) = 5/2·16 = 40. Correct answer is 40.
What is the 5th term of the geometric sequence 2, 6, 18, 54, …? — 162.r = 3. a₅ = 2·3⁴ = 2·81 = 162.
An infinite geometric series converges when: — |r| < 1.The infinite geometric series S∞ = a₁/(1−r) only converges (has a finite sum) when |r| < 1.
What does Σ notation represent? — Sum of terms.The summation symbol Σ (sigma) denotes the sum of a sequence of terms indexed from the lower to upper bound.
Evaluate Σᵢ₌₁⁵ i (i.e., 1 + 2 + 3 + 4 + 5). — 15.Using the formula n(n+1)/2 = 5·6/2 = 15. Or simply 1+2+3+4+5 = 15.
In polar coordinates, what does r represent? — The distance from the origin.In polar coordinates (r, θ), r is the radial distance (magnitude) from the origin to the point.
What is the rectangular equivalent of the polar point (2, 0)? — (2, 0).x = 2·cos(0) = 2, y = 2·sin(0) = 0. Rectangular: (2, 0).
In parametric equations, x = f(t) and y = g(t). What is t called? — The parameter.t is called the parameter — a third independent variable that determines both x and y values as it varies.
The equation r = 5 in polar coordinates describes: — A circle of radius 5 centered at the origin.r = constant means every point is the same distance from the origin — this is a circle of radius 5.
For parametric equations x = cos(t), y = sin(t), eliminating t gives: — x² + y² = 1.Since cos²t + sin²t = 1, substituting x = cos t and y = sin t gives x² + y² = 1 — the unit circle.
What is the formula for the parametric slope dy/dx? — (dy/dt)/(dx/dt).By the chain rule: dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.
A cardioid has the polar equation r = a(1 + cos θ). At θ = π, what is r? — 0.r = a(1 + cos π) = a(1 + (−1)) = a·0 = 0. The cardioid passes through the origin at θ = π.
What distinguishes a vector from a scalar? — Vectors have both magnitude and direction.A vector has both magnitude (size) and direction, while a scalar has only magnitude (e.g., temperature, mass).
What is the magnitude of the vector ⟨3, 4⟩? — 5.|v⃗| = √(3²+4²) = √(9+16) = √25 = 5.
If v⃗ = ⟨a, b⟩, the unit vector û in the same direction is: — ⟨a/|v⃗|, b/|v⃗|⟩.A unit vector has magnitude 1. Divide each component by |v⃗|: û = ⟨a/|v⃗|, b/|v⃗|⟩.
What is 3·⟨2, −4⟩? — ⟨6, −12⟩.Scalar multiplication: 3·⟨2, −4⟩ = ⟨3·2, 3·(−4)⟩ = ⟨6, −12⟩.
Two vectors are perpendicular if their dot product is: — 0.u⃗ · v⃗ = |u⃗||v⃗|cos θ = 0 when cos θ = 0 (θ = 90°), meaning the vectors are perpendicular (orthogonal).
What does the dot product u⃗ · v⃗ equal geometrically? — |u⃗||v⃗|cos θ.The dot product equals |u⃗||v⃗|cos θ, where θ is the angle between the two vectors.
What is the standard unit vector î in 2D? — ⟨1, 0⟩.î = ⟨1, 0⟩ is the standard unit vector along the positive x-axis. ĵ = ⟨0, 1⟩ is along the y-axis.