Pre-Calculus
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Sequences & Series

🔭 Pre-Calculus
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Sequences & Series

Analyze arithmetic and geometric sequences and evaluate finite and infinite series.

A sequence is an ordered list of numbers following a rule. An arithmetic sequence adds a constant difference d each term; a geometric sequence multiplies by a constant ratio r. A series is the sum of the terms of a sequence.

Arithmetic: an=a1+(n−1)dSn=n2(a1+an)\text{Arithmetic: } a_n = a_1 + (n-1)d \qquad S_n = \frac{n}{2}(a_1 + a_n)
Geometric: an=a1⋅rn−1Sn=a1⋅1−rn1−r(r≠1)\text{Geometric: } a_n = a_1 \cdot r^{n-1} \qquad S_n = a_1\cdot\frac{1-r^n}{1-r} \quad (r \neq 1)
Infinite geometric series (∣r∣<1):S∞=a11−r\text{Infinite geometric series } (|r| < 1): \quad S_\infty = \frac{a_1}{1-r}
Summation notation: ∑i=1ni=n(n+1)2∑i=1ni2=n(n+1)(2n+1)6\text{Summation notation: } \sum_{i=1}^{n} i = \frac{n(n+1)}{2} \qquad \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}
📊Infinite Geometric Series
Find the sum: 4 + 2 + 1 + 1/2 + … Here a₁ = 4 and r = 1/2. Since |r| < 1: S∞ = 4/(1 − 1/2) = 4/(1/2) = 8.

Arithmetic Sequence

Constant difference da_n = a₁ + (n−1)dS_n = n/2 · (a₁ + aₙ)Example: 3, 7, 11, 15, … (d=4)

Geometric Sequence

Constant ratio ra_n = a₁ · r^(n−1)S_n = a₁(1−rⁿ)/(1−r)Example: 2, 6, 18, 54, … (r=3)

Arithmetic sequences have constant first differences; geometric sequences have constant ratios.

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Remember This!

To determine whether a sequence is arithmetic or geometric: check first differences (arithmetic) or ratios of consecutive terms (geometric). A sequence can be neither.

✏️ Try It!

What is the sum of the infinite geometric series 6 + 2 + 2/3 + 2/9 + …?

Practice with CalcVerse
→Limit Simulator∑Sequence & Series
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