A sequence is a set of objects (usually numbers) listed in order.

An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted as $d$.

General term of an arithmetic sequence : Page 22

List the first $5$ terms of arithmetic sequence with $a=6$ and $d=-2$.

• $T_1=a+(1-1)(-2)=a=6$
• $T_2=6+(2-1)(-2)=4$
• $T_3=6+(3-1)(-2)=2$
• $T_4=6+(4-1)(-2)=0$
• $T_5=6+(5-1)(-2)=-2$

If $6,p, q,32$ are the first four consecutive terms in an arithmetic sequence, find $p$ and $q$, and then find the general term.

• $T_1=a+(1-1)(d)=a=6$
• $T_4=6+(4-1)(d)=32$

Once $a$ and $d$ are known, we can find any term, $T_n$.

• $p=T_2=(6+\tfrac{26}{3})=\tfrac{44}{3}$

• $q=T_3=(\tfrac{44}{3}+\tfrac{26}{3})=\tfrac{70}{3}$ The general term :

• $T_n=a+(n-1)d=6+(n-1)\tfrac{26}{3}$

Which term is $249$ in the arithmetic sequence $2,15,28,...$ ?

First let's determine the common difference, which can be found by taking any term subtracted by the term before it, in general :

• $T_3-T_2=28-15=13=d$

Now we are looking for the position of the term $249$, so $n$ is the unknown variable :

Let $T_n=249$. Use the general term formula :

Determine if the general term $T_n=n^2+3n$ forms an arithmetic sequence.

We need $T_n$ (given to us already) and $T_{n-1}$, so we can just sub in $n-1$ for $n$.

Now determine if $T_n-T_{n-1}$ is constant :

$2n+2$ is not constant, so the sequence formed from $T_n$ is not arithmetic.