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.

10 min TEKS 8A,8B,8C Algebra 2

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

S_n = n(a₁ + a_n) / 2 n = number of terms, a₁ = first term, a_n = 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

Practice

Sum an arithmetic series

Find the sum of 2 + 5 + 8 + ... + 32.

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Geometric series

S_n = 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.

Practice

Sum a geometric series

For 3 + 6 + 12 + ... + 192, find the sum.

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The four conic sections

The four shapes. Read the equation: + for ellipse / circle, − for hyperbola, only one variable squared means parabola.

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)

Practice

Identify a circle from its equation

The equation x² + y² = 25 represents:

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Practice

Standard form of an ellipse

What is the standard form of an ellipse centered at the origin with horizontal major axis?

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3-second recap

Arithmetic sum: n(a₁ + a_n)/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.