Sequences and Progressions

The One Thing

A sequence is an ordered list generated by a rule. The principal task is to distinguish an arithmetic progression, which has a constant difference, from a geometric progression, which has a constant ratio, and then apply the appropriate formula or recursive rule.

Definitions and Results

• Sequence: An ordered list of numbers written according to a specific rule, such as $2,4,6,8,\dots$. The ordering and rule both matter; the same numbers arranged differently may form a different sequence.
• Term: Each individual number in a sequence. The $n$th term is commonly written as $a_n$.
• Pattern: A regular relationship or rule used to generate the terms of a sequence.
• Arithmetic Progression (AP): A sequence in which the difference between consecutive terms is constant.
• Common Difference: The fixed difference between consecutive terms of an AP, usually denoted by $d$:

$$d=a_{n+1}-a_n.$$ A sequence is an AP only if this difference is the same for every pair of consecutive terms. The value of $d$ may be positive, negative, or zero: - $d>0$: increasing progression; - $d<0$: decreasing progression; - $d=0$: constant terms.

• First Term: The initial term of a sequence or progression, usually denoted by $a$.
• General Term of an AP: For an AP with first term $a$, common difference $d$, and term number $n$,

$$a_n=a+(n-1)d.$$

• Sum of the First $n$ Terms of an AP:

$$S_n=\frac{n}{2}[2a+(n-1)d].$$ If the last term is $l$, the equivalent formula is $$S_n=\frac{n}{2}(a+l).$$

• Geometric Progression (GP): A sequence in which each term is obtained by multiplying the previous term by a constant non-zero number.
• Common Ratio: The fixed ratio of a term to the preceding term in a GP, usually denoted by $r$:

$$r=\frac{a_{n+1}}{a_n},$$ provided the preceding term is not zero.

• General Term of a GP: For a GP with first term $a$, common ratio $r$, and term number $n$,

$$a_n=a{r}^{n-1}.$$ The behaviour depends on $r$: - $r>1$ generally increases positive terms; - $0<r<1$ generally decreases positive terms; - $r<0$ causes alternating signs.

• Recursive Rule: A rule defining a term using one or more preceding terms. For example,

$$a_n=a_{n-1}+3.$$ A recursive AP rule is $$a_1=a,a_n=a_{n-1}+d\text{for }n>1.$$ A recursive GP rule is $$a_1=a,a_n=r{a}_{n-1}\text{for }n>1.$$

• Finite Sequence: A sequence with a limited number of terms.
• Infinite Sequence: A sequence that continues without an ending term.

Worked Methods

1. Identifying a pattern

• Compare consecutive terms in their given order.
• Calculate differences to test for an AP.
• Calculate ratios, where defined, to test for a GP.
• Check several consecutive terms before concluding that a rule is present.
• Preserve the original order, since changing the order can change the sequence and its progression properties.

For example, the sequence $$2,4,6,8,\dots$$ has consecutive differences $$4-2=2,6-4=2,8-6=2.$$ It is therefore an AP with common difference $d=2$.

2. Finding a term of an AP

• Identify the first term $a$.
• Identify the common difference $d$.
• Identify the required term number $n$.
• Substitute into

$$a_n=a+(n-1)d.$$

• Simplify the result.

The formula is valid only after the correct first term, common difference, and term number have been identified.

3. Finding the sum of an AP

• Identify $n$, the number of terms.
• Identify the first term $a$.
• Identify the common difference $d$, and use

$$S_n=\frac{n}{2}[2a+(n-1)d],$$ or identify the last term $l$ and use $$S_n=\frac{n}{2}(a+l).$$

• Substitute the known values.
• Simplify the expression.

4. Finding a term of a GP

• Identify the first term $a$.
• Identify the non-zero common ratio $r$ by comparing consecutive terms:

$$r=\frac{a_{n+1}}{a_n}.$$

• Identify the required term number $n$.
• Substitute into

$$a_n=a{r}^{n-1}.$$

• Simplify the result.

The ratio must be constant across the relevant consecutive terms, and the preceding term used in the ratio must not be zero.

5. Using recursive rules

For an AP:

• Set the first term:

$$a_1=a.$$

• Add the common difference to obtain each subsequent term:

$$a_n=a_{n-1}+d\text{for }n>1.$$

For a GP:

• Set the first term:

$$a_1=a.$$

• Multiply each term by the common ratio:

$$a_n=r{a}_{n-1}\text{for }n>1.$$

Recursive rules generate terms step by step, whereas explicit formulas find a chosen term directly. Both describe the same progression when their initial value and difference or ratio agree.

6. Applying progressions to situations

Progressions can model regular savings, repeated growth or decay, seating arrangements, and patterns in objects. The modelling process is to identify the ordered quantities, determine whether the change is a constant difference or ratio, and then select the corresponding AP or GP rule.

Where It Goes Wrong

• Applying the AP formula before checking that all consecutive differences are equal.
• Applying the GP formula before checking that all consecutive ratios are equal.
• Forgetting that the position of a term matters; changing the order can change the sequence and its progression properties.
• Using an incorrect first term, common difference, common ratio, or term number in the formula.
• Calculating a common ratio using a preceding term that is zero, although

$$r=\frac{a_{n+1}}{a_n}$$ requires a non-zero preceding term.

• Forgetting the behaviour associated with the sign and size of the parameter: $d$ may be positive, negative, or zero, while $r<0$ produces alternating signs.

What Gets Asked

This material supports questions that require students to:

• define a sequence, term, pattern, AP, GP, common difference, common ratio, finite sequence, or infinite sequence;
• determine whether a given ordered list is an AP or GP;
• calculate a common difference using

$$d=a_{n+1}-a_n;$$

• calculate a common ratio using

$$r=\frac{a_{n+1}}{a_n};$$

• find the $n$th term of an AP using

$$a_n=a+(n-1)d;$$

• find the $n$th term of a GP using

$$a_n=a{r}^{n-1};$$

• calculate the sum of the first $n$ terms of an AP using either

$$S_n=\frac{n}{2}[2a+(n-1)d]$$ or $$S_n=\frac{n}{2}(a+l);$$

• generate terms from the recursive AP rule

$$a_1=a,a_n=a_{n-1}+d$$ or the recursive GP rule $$a_1=a,a_n=r{a}_{n-1};$$

• convert between recursive and explicit descriptions of a progression;
• explain how positive, negative, or zero $d$, and positive or negative values of $r$, affect the progression;
• model regular savings, repeated growth or decay, seating arrangements, and patterns in objects using sequences or progressions.