Sequences and Series

The One Thing

Sequences list numbers according to a rule, whereas series add the terms of a sequence. Arithmetic progressions and geometric progressions are identified by constant differences and ratios respectively, and their term and sum formulas provide efficient methods for calculation; the A.M.–G.M. inequality extends these ideas to inequalities and optimization.

Definitions and Results

• Sequence: An ordered list of numbers written as $a_1,a_2,a_3,\dots$ according to a rule.
• Term: Each individual number in a sequence; the $n$th term is denoted by $a_n$.
• Series: The sum of the terms of a sequence, such as $a_1+a_2+a_3+\cdots$.
• Arithmetic Progression (A.P.): A sequence in which the difference between consecutive terms is constant.
• Common Difference: The constant difference $d$ between consecutive terms of an A.P.:

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

• Nth term of an A.P.: If the first term is $a$ and the common difference is $d$, then

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

• Sum of the first $n$ terms of an A.P.:

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

• Equidistant-term property of an A.P.: Terms equally distant from the beginning and end have a constant sum:

$$a_1+a_n=a_2+a_{n-1}=\cdots .$$

• Three terms in A.P.: Three numbers $x,y,z$ are in A.P. if

$$2y=x+z.$$

• Geometric Progression (G.P.): A sequence in which the ratio of each term to the preceding term is constant.
• Common Ratio: The constant ratio $r$ between consecutive non-zero terms of a G.P.:

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

• Nth term of a G.P.: If the first term is $a$ and the common ratio is $r$, then

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

• Sum of the first $n$ terms of a G.P.: For $r\ne 1$,

$$S_n=a\frac{r^n-1}{r-1}$$ or, equivalently, $$S_n=a\frac{1-r^n}{1-r}.$$ If $r=1$, then $$S_n=na.$$

• Infinite G.P.: A geometric series continuing without end. Its sum to infinity is finite only when $|r|<1$.
• Sum to infinity of a G.P.: Provided $|r|<1$,

$$S_{\infty }=\frac{a}{1-r}.$$ If $|r|\ge 1$, the usual infinite G.P. does not have a finite sum.

• Three terms in G.P.: Three positive numbers $x,y,z$ are in G.P. if

$$y^2=xz.$$

• Insertion of arithmetic means: To insert $n$ arithmetic means between $a$ and $b$, use

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

• Insertion of geometric means: To insert $n$ geometric means between positive numbers $a$ and $b$, use

$$r={\left(\frac{b}{a}\right)}^{1/(n+1)}.$$

• Arithmetic Mean (A.M.): For two numbers $a$ and $b$,

$$\text{A.M.}=\frac{a+b}{2}.$$

• Geometric Mean (G.M.): For two positive numbers $a$ and $b$,

$$\text{G.M.}=\sqrt{ab}.$$

• A.M.–G.M. Inequality: For positive numbers $a$ and $b$,

$$\frac{a+b}{2}\ge \sqrt{ab},$$ with equality when $a=b$.

• A.M.–G.M. inequality for $n$ positive numbers: For positive real numbers $x_1,x_2,\dots ,x_n$,

$$\frac{x_1+x_2+\cdots +x_n}{n}\ge (x_1{x}_2\cdots x_n{)}^{1/n}.$$ Equality occurs only when all the numbers are equal.

• Finite Series: A series containing a fixed, finite number of terms.
• Explicit sequence: A sequence may be defined directly, for example,

$$a_n=2n+1.$$

• Recursive sequence: A sequence may be defined by specifying each term using earlier terms.

Worked Methods

Finding a term in an A.P.

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

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

• Simplify to obtain the required term.

The linear form arises because consecutive terms differ by the same constant amount.

Finding the sum of an A.P.

• Identify $a$, $d$, and $n$.
• Use

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

• If the last term $l$ is known instead of $d$, use

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

• Alternatively, pair terms equally distant from the beginning and end, using

$$a_1+a_n=a_2+a_{n-1}=\cdots .$$

Identifying three terms in an A.P.

• Let the three terms be $x,y,z$, in that order.
• Test the condition

$$2y=x+z.$$

• If the equality holds, the three numbers are in A.P.

Inserting arithmetic means

• Count the number $n$ of means to be inserted between $a$ and $b$.
• There are $n+1$ equal intervals.
• Calculate

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

• Starting from $a$, add $d$ successively to obtain the inserted arithmetic means and finally $b$.

Finding a term in a G.P.

• Identify the first term $a$, common ratio $r$, and position $n$.
• Substitute into

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

• Simplify the resulting power expression.

The exponential form arises because each term is obtained by multiplying the preceding term by the same ratio.

Finding the sum of a finite G.P.

• Identify $a$, $r$, and $n$.
• If $r\ne 1$, use

$$S_n=a\frac{r^n-1}{r-1}$$ or $$S_n=a\frac{1-r^n}{1-r}.$$

• If $r=1$, use

$$S_n=na.$$

Finding the sum to infinity of a G.P.

• Confirm that the series is geometric.
• Check the necessary condition

$$|r|<1.$$

• If this condition holds, use

$$S_{\infty }=\frac{a}{1-r}.$$

• If $|r|\ge 1$, conclude that the usual infinite G.P. has no finite sum.

Identifying three terms in a G.P.

• Let the three positive terms be $x,y,z$, in that order.
• Test the condition

$$y^2=xz.$$

• If the equality holds, the numbers are in G.P.

Inserting geometric means

• Confirm that the endpoint numbers $a$ and $b$ are positive.
• Count the number $n$ of means required.
• Calculate

$$r={\left(\frac{b}{a}\right)}^{1/(n+1)}.$$

• Starting from $a$, multiply successively by $r$ to obtain the inserted geometric means and finally $b$.

Applying the A.M.–G.M. inequality

• Confirm that all quantities are positive.
• For two numbers, compare

$$\frac{a+b}{2}\text{and}\sqrt{ab}.$$

• For $n$ numbers, compare

$$\frac{x_1+x_2+\cdots +x_n}{n}\text{and}(x_1{x}_2\cdots x_n{)}^{1/n}.$$

• Use equality only when the relevant quantities are equal.
• In optimization, use the equality condition to identify the extremal case: equal positive quantities give the greatest product for a fixed sum and the smallest sum for a fixed product.

Where It Goes Wrong

• Confusing a sequence with a series: finding a term and finding a sum are different tasks.
• Applying an A.P. formula without first verifying a constant difference, or applying a G.P. formula without verifying a constant ratio.
• Using the finite G.P. formula when $r=1$, instead of using $S_n=na$.
• Applying the sum-to-infinity formula without checking $|r|<1$; when $|r|\ge 1$, there is no finite sum.
• Forgetting the positivity condition for geometric means and for the A.M.–G.M. inequality.
• Claiming equality in the A.M.–G.M. inequality without requiring $a=b$, or, for several numbers, without requiring all the numbers to be equal.

What Gets Asked

• Define a sequence, term, series, finite series, A.P., G.P., common difference, and common ratio.
• Determine a specified term of an A.P. or G.P. using $a_n=a+(n-1)d$ or $a_n=a{r}^{n-1}$.
• Find the sum of the first $n$ terms of an A.P. or G.P.
• Find an A.P. sum using the first and last terms.
• Determine whether three numbers are in A.P. using $2y=x+z$, or in G.P. using $y^2=xz$.
• Insert arithmetic means or geometric means between two given numbers.
• Find the sum to infinity of an infinite G.P. and state the condition $|r|<1$.
• Distinguish explicitly defined sequences, such as $a_n=2n+1$, from recursively defined sequences.
• State and apply the A.M.–G.M. inequality for two or $n$ positive numbers.
• Use equality in the A.M.–G.M. relation to prove inequalities, find minimum values, and determine when a product or sum is optimized.