4.2 Arithmetic Sequence and General Term

Definition

An arithmetic sequence (or arithmetic progression, A.P.) is a sequence in which each term after the first is obtained by adding a fixed constant called the common difference $d$ to the preceding term.

$a_1,a_1+d,a_1+2d,a_1+3d,\dots$

The common difference is found by: $d=a_{n+1}-a_n\text{(constant for all }n\text{)}$

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General Term (nth Term)

The general term (or $n$th term) of an arithmetic sequence is:

$a_n=a_1+(n-1)d$

where:

• $a_1$ = first term
• $d$ = common difference
• $n$ = position of the term

Example: Find the 15th term of $4,9,14,19,\dots$

$a_1=4,d=9-4=5$ $a_{15}=4+(15-1)(5)=4+70=74$

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Finding $a_1$ and $d$ from Two Non-Consecutive Terms

If two terms $a_m$ and $a_n$ are known, set up a system of equations:

$a_m=a_1+(m-1)d$ $a_n=a_1+(n-1)d$

Subtract one equation from the other to find $d$, then substitute back to find $a_1$.

Example: If $a_3=7$ and $a_9=25$, find $a_1$ and $d$.

$a_3=a_1+2d=7\cdots (1)$ $a_9=a_1+8d=25\cdots (2)$

Subtracting (1) from (2): $6d=18\Rightarrow d=3$

Substituting into (1): $a_1=7-2(3)=1$

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Three Consecutive Terms in A.P.

If $x$, $y$, $z$ are three consecutive terms of an A.P., then:

$y-x=z-y⟹2y=x+z$

This means the middle term is the arithmetic mean of the outer two terms.

Tip: When three numbers in A.P. are unknown, let them be $a-d$, $a$, $a+d$ to simplify calculations.

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Arithmetic Mean

The arithmetic mean $A$ between two numbers $a$ and $b$ is:

$A=\frac{a+b}{2}$

This works because $a$, $A$, $b$ must form an A.P., so $A-a=b-A$, giving $2A=a+b$.

Example: The arithmetic mean between $\sqrt{2}$ and $3\sqrt{2}$ is: $A=\frac{\sqrt{2}+3\sqrt{2}}{2}=\frac{4\sqrt{2}}{2}=2\sqrt{2}$

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Finding the Position of a Term

To find which position $n$ a value $X$ occupies in an A.P.:

1. Set $a_n=X$: $a_1+(n-1)d=X$
2. Solve for $n$
3. Check: $n$ must be a positive integer; if not, $X$ is not a term of the sequence.

Example: Is $101$ a term of $5,8,11,14,\dots$?

$5+(n-1)(3)=101⟹3(n-1)=96⟹n-1=32⟹n=33$

Since $n=33$ is a positive integer, $101$ is the 33rd term.

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Key Formulas Summary

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