Study sheet: Sequences and Recurrence Relations

📋 Course Outline

1. Sequence Notation and Recurrences
2. Arithmetic Sequences
3. Arithmetic Sums
4. Geometric Sequences
5. Applications of Geometric Growth

📖 1. Sequence Notation and Recurrences

★ Must-know

📌 A sequence can be defined explicitly by a formula such as $u_n=\frac{n}{n+1}$ or recursively by giving initial term(s) and a relation connecting consecutive terms.

• To calculate successive terms of a recursively defined sequence, start from the given initial term(s), substitute them into the recurrence, and repeat the relation until the requested index is reached.

• A recurrence model translates a verbal evolution into algebra by identifying the previous value, the next value, and any multiplication, addition, percentage change, or averaging operation between them.

Further detail

• The recurrence $u_{n+1}=2u_n-1$ with $u_0=5$ defines a sequence whose first terms are obtained successively from the preceding term.

💡 Memory Hook

Explicit formula gives a term directly, whereas recurrence builds it from previous terms.

📖 2. Arithmetic Sequences

🔑 Key Concepts & Definitions

• Arithmetic sequence : a sequence for which the same number r, called the common difference, is added to each term to obtain the next term.

★ Must-know

📐 Formula — An arithmetic sequence satisfies $u_{n+1}=u_n+r$ and, with first term $u_0$, has explicit form $u_n=u_0+rn$ for every natural number n.

Further detail

• If the renewable-energy share is 5.7% in 2025 and increases by 0.7 percentage points each year, then $u_0=5.7$, $r=0.7$, and $u_n=5.7+0.7n$ for the share in year 2025+n.

📐 Formula — For an arithmetic sequence of common difference r, any two terms satisfy $u_n=u_k+(n-k)r$.

💡 Memory Hook

A staircase rising by equal vertical steps of size r.

📖 3. Arithmetic Sums

★ Must-know

📐 Formula — The sum of the integers from 1 to n is $\sum_{i=1}^{n}i=1+2+\cdots+n=\frac{n(n+1)}{2}$.

• To compute a finite sum written with sigma notation, identify the lower index, upper index, and general term, then substitute the required indices or apply the appropriate summation formula.

Further detail

• A sum such as $\sum_{k=5}^{12}k$ represents the consecutive integers from 5 through 12, inclusive.

• An algorithm for calculating a sum such as $S_n=\sum_{k=1}^{n}\frac{1}{k}$ repeatedly adds the term corresponding to each index from 1 to n.

💡 Memory Hook

Gauss: pair the first and last terms.

📖 4. Geometric Sequences

🔑 Key Concepts & Definitions

• Geometric sequence : a sequence for which the same number q, called the common ratio, multiplies each term to obtain the next term.

★ Must-know

📐 Formula — The sum of the first n+1 powers of q is $\sum_{i=0}^{n}q^i=1+q+q^2+\cdots+q^n=\frac{1-q^{n+1}}{1-q}$ when $q\ne1$.

📐 Formula — For a geometric sequence, the sum of its first n+1 terms is $v_0+v_1+\cdots+v_n=v_0\frac{1-q^{n+1}}{1-q}$ when $q\ne1$.

Further detail

• If the renewable-energy share is 5.7% in 2025 and grows by 5.5% per year, then $v_0=5.7$, $q=1.055$, and $v_{n+1}=1.055v_n$.

💡 Memory Hook

A snowball repeatedly multiplied by the same ratio q.

📖 5. Applications of Geometric Growth

★ Must-know

• To determine when a geometric quantity reaches a target, express the target condition with the model $u_n=u_0q^n$ and solve or inspect the resulting equation for the index n.

📐 Formula — For the glass-recycling model, the amount recycled in year 2025+n is $u_n=200\times1.10^n$.

Further detail

• A deposit of 2000 euros earning 5% compound annual interest is modeled after n years by $u_n=2000\times1.05^n$.

• If 200 tonnes of glass are recycled in 2025 and the amount increases by 10% each year, then the sequence has initial term $u_0=200$ and common ratio $q=1.10$.

• The total glass recycled from 2025 through 2033 inclusive is obtained by summing the geometric terms $u_0+u_1+\cdots+u_8$.

💡 Memory Hook

Repeated percentage growth → geometric sequence → power or geometric-sum model.

📊 Synthesis Tables

Arithmetic and Geometric Sequences

Feature	Arithmetic sequence	Geometric sequence
Change from one term to the next	Add r	Multiply by q
Recurrence	$u_{n+1}=u_n+r$	$u_{n+1}=qu_n$
Explicit form	$u_n=u_0+rn$	$u_n=u_0q^n$
Typical percentage model	Not constant percentage growth	Constant percentage growth