Find the 8th term of the arithmetic sequence: a₁ = 7, d = −2. — −7.a₈ = 7 + (8−1)(−2) = 7 − 14 = −7.
The sum of the first n natural numbers formula is: — n(n+1)/2.Σᵢ₌₁ⁿ i = n(n+1)/2. This is Gauss's famous formula for the sum of consecutive integers.
For the geometric series 3 + 6 + 12 + 24, what is S₆? — 189.a₁=3, r=2. S₆ = 3·(1−2⁶)/(1−2) = 3·(1−64)/(−1) = 3·63 = 189.
What is the common ratio of the geometric sequence 81, 27, 9, 3, …? — 1/3.r = 27/81 = 1/3. Each term is divided by 3.
Find S∞ for the series 12 + 4 + 4/3 + 4/9 + … — 18.a₁=12, r=1/3. S∞ = 12/(1−1/3) = 12/(2/3) = 18.
A repeating decimal 0.333… = 0.3̄ can be expressed as a fraction using: — Infinite geometric series with r=1/10.0.333… = 3/10 + 3/100 + … This is a geometric series with a₁=3/10, r=1/10. Sum = (3/10)/(1−1/10) = (3/10)/(9/10) = 1/3.
Which sequence is neither arithmetic nor geometric? — 1, 1, 2, 3, 5, 8.The Fibonacci sequence 1, 1, 2, 3, 5, 8 has no constant difference or ratio.
Find the number of terms in the arithmetic sequence 5, 9, 13, …, 101. — 25.aₙ = 5 + (n−1)·4 = 101 → (n−1)·4 = 96 → n−1 = 24 → n = 25.
Insert two arithmetic means between 4 and 22 (find them). — 10 and 16.The sequence is 4, a₂, a₃, 22 — four terms with a₁=4, a₄=22. d = (22−4)/3 = 6. So terms are 4, 10, 16, 22.
Convert the polar point (6, π) to rectangular coordinates. — (−6, 0).x = 6·cos(π) = 6·(−1) = −6. y = 6·sin(π) = 0. Rectangular: (−6, 0).
For the parametric curve x = t², y = t³, find dy/dx in terms of t. — 3t/2.dy/dt = 3t², dx/dt = 2t. So dy/dx = 3t²/(2t) = 3t/2.
Convert the rectangular equation x² + y² = 16 to polar form. — r = 4.Since r² = x² + y², we get r² = 16, so r = 4 (taking r > 0). This is a circle of radius 4.
At what values of t does the parametric curve x = sin t, y = cos(2t) cross the x-axis (y = 0)? — t = π/4, 3π/4.y = cos(2t) = 0 when 2t = π/2 + nπ, i.e. t = π/4, 3π/4, 5π/4, 7π/4.
What is the rectangular form of r = 2cos θ? — (x−1)² + y² = 1.Multiply both sides by r: r² = 2r cos θ → x²+y² = 2x → (x−1)²+y² = 1. A circle of radius 1 centered at (1, 0).
Eliminate the parameter from x = 2t − 1 and y = t + 3. — y = (x+7)/2.From x = 2t−1: t = (x+1)/2. Substitute: y = (x+1)/2 + 3 = (x+1+6)/2 = (x+7)/2.
The polar curve r = a(1 − cos θ) is called a: — Cardioid.r = a(1 − cos θ) is a cardioid — a heart-shaped curve symmetric about the x-axis passing through the origin.
For the curve r = 3sin(2θ) (a 4-petal rose), how many petals does it have? — 4.r = a·sin(nθ) gives 2n petals when n is even. For n=2: 4 petals.
Find the slope of the tangent to x = t², y = 2t at t = 3. — 1/3.dy/dt = 2, dx/dt = 2t = 6 at t=3. Slope = dy/dx = 2/6 = 1/3.
Find the unit vector in the direction of ⟨3, 4⟩. — ⟨3/5, 4/5⟩.|v⃗| = 5. Unit vector = ⟨3/5, 4/5⟩.
Find the angle between ⟨1, 0⟩ and ⟨0, 1⟩. — 90°.Dot product = 0. cos θ = 0/(1·1) = 0. θ = 90°. These are the standard x and y unit vectors, which are perpendicular.
Are ⟨3, −2⟩ and ⟨4, 6⟩ perpendicular? — Yes, because their dot product is 0.Dot product = 3·4 + (−2)·6 = 12 − 12 = 0. Since the dot product is 0, the vectors are perpendicular.
A vector v⃗ has magnitude 10 and makes a 30° angle with the x-axis. Find its components. — ⟨5√3, 5⟩.vₓ = 10cos30° = 10·(√3/2) = 5√3. v_y = 10sin30° = 10·(1/2) = 5. So v⃗ = ⟨5√3, 5⟩.