Arithmetic Progressions

The One Thing

An arithmetic progression (AP) is determined by its first term $a$ and constant common difference $d$. Its terms and sums can be found efficiently using $$a_n=a+(n-1)d$$ and $$S_n=\frac{n}{2}[2a+(n-1)d]=\frac{n}{2}(a+l).$$

Definitions and Results

• Sequence: An ordered list of numbers arranged according to a rule.

• Arithmetic Progression (AP): A sequence in which the difference between every pair of consecutive terms is constant. It has the form

$$a,a+d,a+2d,a+3d,\dots$$

• First term $(a)$: The first number in an arithmetic progression.

• Common difference $(d)$: The constant difference between consecutive terms:

$$d=\text{second term}-\text{first term}$$ or, specifically, $$d=a_2-a_1=a_3-a_2.$$ This difference must remain the same throughout the progression.

• General or nth term $(a_n)$: The term at the $n$th position of an AP:

$$a_n=a+(n-1)d.$$ This follows because $d$ is added exactly $n-1$ times to the first term.

• Last term $(l)$: The final term when a finite number of terms of an AP is considered:

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

• Number of terms $(n)$: The total number of terms included in a sequence or sum.

• Sum of the first $n$ terms $(S_n)$:

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

• Increasing AP: An AP for which $d>0$; the terms increase.

• Decreasing AP: An AP for which $d<0$; the terms decrease.

• Constant AP: An AP for which $d=0$; all terms are equal.

• Number of terms when $a$, $l$, and $d$ are known:

$$n=\frac{l-a}{d}+1,$$ provided $d\ne 0$.

• Average of the terms: The average of the first and last terms is

$$\frac{a+l}{2}.$$ This is also the average of the first $n$ terms of an AP.

• Middle term of an odd-length AP: If an AP has an odd number of terms, its middle term equals the average of the first and last terms.

Worked Methods

1. Identifying an arithmetic progression

• Subtract each term from the following term.
• Check that the differences are equal.
• Identify the first term $a$ and common difference $d$.
• If the differences are constant, represent the AP as

$$a,a+d,a+2d,a+3d,\dots$$

For example, in $$5,8,11,14,\dots$$ the first term is $$a=5$$ and the common difference is $$d=8-5=3.$$

2. Finding the nth term

• Identify the first term $a$.
• Identify the common difference $d$.
• Substitute these values and the required position $n$ into

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

• Simplify the expression if required.

For $$5,8,11,14,\dots ,$$ where $a=5$ and $d=3$, $$a_n=5+3(n-1).$$ Therefore, $$a_n=5+3n-3=3n+2.$$

3. Finding the last term

• Identify $a$, $d$, and $n$.
• Use the nth-term formula:

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

• Substitute the known values and simplify.

The last term is the $n$th term, so the same formula applies.

4. Finding the sum using the first term and common difference

• Identify $a$, $d$, and $n$.
• Use

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

• Substitute the values.
• Simplify to obtain the total.

For the first 10 terms of $$2,5,8,\dots ,$$ the values are $$a=2,d=3,n=10.$$ Thus, $$S_{10}=\frac{10}{2}[2(2)+9(3)]$$ and $$S_{10}=155.$$

5. Finding the sum when the last term is known

• Identify the first term $a$, last term $l$, and number of terms $n$.
• Pair the first and last terms, the second and second-last terms, and so on.
• Each pair has the same sum, $a+l$.
• Use

$$S_n=\frac{n}{2}(a+l).$$

This pairing method explains why the sum formula works: the first and last terms, the second and second-last terms, and subsequent pairs all have equal sums.

6. Finding the number of terms

• Identify $a$, $l$, and $d$.
• Use

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

• Rearrange:

$$n=\frac{l-a}{d}+1.$$

• This formula requires $d\ne 0$.

7. Solving word problems involving APs

• Determine the first term $a$.
• Determine the common difference $d$.
• Determine the number of terms $n$, if given or required.
• Determine the last term $l$, when applicable.
• Select the appropriate nth-term or sum formula.
• Substitute the values and interpret the result in the context of the problem.

Arithmetic progressions can model regularly increasing or decreasing quantities, including seating arrangements, savings, and equally spaced patterns.

Where It Goes Wrong

• The common difference must be constant throughout the progression; checking only one pair of consecutive terms is insufficient.
• The nth term is

$$a+(n-1)d,$$ not $a+nd$; the difference is added $n-1$ times.

• When using

$$S_n=\frac{n}{2}[2a+(n-1)d],$$ the factor $n-1$ must be retained.

• The formula

$$n=\frac{l-a}{d}+1$$ requires $d\ne 0$; it cannot be applied by division when the AP is constant.

• If $d$ is negative, the same formulas remain valid, but signs must be handled correctly.
• In word problems, $a$, $d$, $n$, and $l$ must be identified carefully before choosing a formula.

What Gets Asked

• Determine whether a sequence is an arithmetic progression.
• Find the first term and common difference.
• Find a particular term using

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

• Derive or use the nth term for a sequence such as

$$5,8,11,14,\dots$$ to obtain $$a_n=3n+2.$$

• Find the last term when $a$, $d$, and $n$ are known.
• Find the sum of the first $n$ terms using

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

• Find a sum when the first and last terms are known using

$$S_n=\frac{n}{2}(a+l).$$

• Find the number of terms when the first term, last term, and common difference are given.
• Use the middle-term and average properties of an AP.
• Apply AP formulas to regularly increasing or decreasing contexts, including seating arrangements, savings, and equally spaced patterns.