Sequences, Series & Conic Sections
Algebra 1 introduced arithmetic and geometric sequences. Algebra 2 sums them with closed-form formulas, and adds the four conic sections — circle, ellipse, parabola, hyperbola — recognized at sight from the equation.
Sums and shapes
This lesson combines two big topics — series sums (compressing infinite addition into one formula) and conic sections (recognizing a curve from its equation). Both reward pattern recognition.
Arithmetic series
Sn = n(a1 + an) / 2 n = number of terms, a1 = first term, an = last term Average of first and last terms, times the count. (Pair them up.)
Example
2 + 5 + 8 + ... + 32 (common difference 3) n = (32 − 2)/3 + 1 = 11 terms Sum = 11 · (2 + 32) / 2 = 11 · 17 = 187
Sum an arithmetic series
Find the sum of the arithmetic series 2 + 5 + 8 + ... + 32.
a₁ = 2, d = 3, last term aₙ = 32. n = (32 − 2)/3 + 1 = 11. Sum = n(a₁ + aₙ)/2 = 11 · 34 / 2 = 187.
Geometric series
Sn = a1(rn − 1) / (r − 1) (r ≠ 1) Infinite series (|r| < 1): S∞ = a1 / (1 − r) Infinite sums converge only when the common ratio's absolute value is less than 1.
Sum a geometric series
For the geometric series 3 + 6 + 12 + ... + 192, what is the sum?
a₁ = 3, r = 2. 192 = 3 · 2ⁿ⁻¹ → 2ⁿ⁻¹ = 64 → n = 7. Sum = a₁(rⁿ − 1)/(r − 1) = 3(128 − 1)/1 = 381.
The four conic sections
Recognition checklist
- x² + y² = r²: circle (radius r)
- x²/a² + y²/b² = 1: ellipse
- x²/a² − y²/b² = 1: hyperbola (note minus sign)
- y = ax² + bx + c: parabola (only one variable squared)
Identify a circle from its equation
The equation x² + y² = 25 represents:
x² + y² = r² is a circle centered at origin. Here r² = 25, so r = 5.
Standard form of an ellipse
What is the standard form of an ellipse centered at the origin with horizontal major axis?
Ellipse centered at origin: x²/a² + y²/b² = 1. Major axis is horizontal when a > b (a is under x²).
3-second recap
- Arithmetic sum: n(a1 + an)/2 — pair up the ends.
- Geometric sum: a1(rn − 1)/(r − 1). Infinite if |r| < 1: a1/(1 − r).
- Conics: + ellipse/circle, − hyperbola, single squared term parabola.
- Read the equation form first; pick formulas after that.
Check yourself
Quick check #1
Find the sum of the first 5 terms of the arithmetic series 2 + 5 + 8 + 11 + 14.
Sₙ = n(a₁ + aₙ)/2. With n = 5, a₁ = 2, a₅ = 14: S₅ = 5(2 + 14)/2 = 5·8 = 40.
Quick check #2
Which conic section has the equation x²/9 + y²/4 = 1?
Standard form x²/a² + y²/b² = 1 with DIFFERENT positive denominators is an ellipse. (If a = b, it's a circle. If signs differ, hyperbola. One squared term + one linear term = parabola.)