Sequences & Series Formulas
IB Mathematics Analysis & Approaches
📐 Arithmetic Sequences
General Form:
A sequence where each term differs from the previous term by a constant value \(d\) (common difference).
\(u_1, u_2, u_3, u_4, \ldots\)
nth Term Formula:
\[u_n = u_1 + (n-1)d\]
where \(u_1\) = first term, \(d\) = common difference, \(n\) = term number
Sum of First n Terms:
\[S_n = \frac{n}{2}\left(2u_1 + (n-1)d\right)\]
\[S_n = \frac{n}{2}(u_1 + u_n)\]
where \(S_n\) = sum of first \(n\) terms, \(u_n\) = nth term
📊 Geometric Sequences
General Form:
A sequence where each term is obtained by multiplying the previous term by a constant value \(r\) (common ratio).
\(a, ar, ar^2, ar^3, \ldots\)
nth Term Formula:
\[u_n = u_1 \cdot r^{n-1}\]
where \(u_1\) = first term, \(r\) = common ratio, \(n\) = term number
Sum of First n Terms (when \(r \neq 1\)):
\[S_n = \frac{u_1(1-r^n)}{1-r}\]
\[S_n = \frac{u_1(r^n-1)}{r-1}\]
Both formulas are equivalent; use the one that best suits the problem
Sum to Infinity (when \(|r| < 1\)):
\[S_{\infty} = \frac{u_1}{1-r}\]
Note: A geometric series converges only when \(|r| < 1\)
Σ Sigma Notation
General Sigma Notation:
Sigma notation provides a concise way to express the sum of a series.
\[\sum_{r=1}^{n} u_r = u_1 + u_2 + u_3 + \cdots + u_n\]
Arithmetic Series in Sigma Notation:
\[\sum_{r=1}^{n} \left(u_1 + (r-1)d\right)\]
Geometric Series in Sigma Notation:
\[\sum_{r=0}^{n-1} u_1 \cdot r^r \quad \text{or} \quad \sum_{r=1}^{n} u_1 \cdot r^{r-1}\]
🔑 Quick Reference
Finding Common Difference (Arithmetic):
\[d = u_{n+1} - u_n\]
Finding Common Ratio (Geometric):
\[r = \frac{u_{n+1}}{u_n}\]
💡 Tip: Always identify whether a sequence is arithmetic (constant difference) or geometric (constant ratio) before applying formulas.