a)For the sequence $7,14,21,28,35,...$, what number is added each time to get the next term?
b)Find a formula for the general term of $7,14,21,28,35,...\qquad u_n=$
c)Find a formula for the general term of $8,15,22,29,36,...\qquad u_n=$
d)Find a formula for the general term of $3,10,17,24,31,...\qquad u_n=$
e)Find a formula for the general term of $50,57,64,71,78,...\qquad u_n=$
f)Find a formula for the general term of $-7,-14,-21,-28,-35...\qquad u_n=$
g)Find a formula for the general term of $-8,-15,-22,-29,-36...\qquad u_n=$
h)Find a formula for the general term of $0,-7,-14,-21,-28,-35...\qquad u_n=$
i)Find a formula for the general term of $18,11,4,-3,-10,...\qquad u_n=$
j)Find a formula for the general term of $-100,-107,-114,-121,-128...\qquad u_n=$
General Term of Arithmetic Sequences
Find the general term, $u_n$, for the following sequences.
*Write your answers in the form $an+b$ where $a,b$ are numbers.
a)$1,3,5,7,9,\ldots \qquad u_n=$
b)$15,18,21,24,27,\ldots \qquad u_n=$
c)$6,17,28,39,50,\ldots \qquad u_n=$
d)$13,5,-3,-11,-19,\ldots \qquad u_n=$
e)$-11,-14,-17,-20,-23,\ldots \qquad u_n=$
f)*Write your answer in the form $\frac{\boxed{\phantom{1}}}{\boxed{\phantom{3}}}n+\frac{\boxed{\phantom{1}}}{\boxed{\phantom{6}}}:$ $\frac{1}{2},\frac{5}{6},\frac{7}{6},\frac{3}{2},\frac{11}{6},\ldots \qquad u_n=$
The $n^{th}$ Term of an Arithmetic Sequence
For arithmetic sequences, the number that is added each time to find the next term is called the
.
It is usually denoted by the letter
.
$$u_1\overset{+d}{\overset{\Huge\frown}{\phantom{A},\phantom{B}}}\underset{u_2}{\text{___}}\overset{+d}{\overset{\Huge\frown}{\phantom{A},\phantom{B}}}\underset{u_3}{\text{___}}\overset{+d}{\overset{\Huge\frown}{\phantom{A},\phantom{B}}}\underset{u_4}{\text{___}}\overset{+d}{\overset{\Huge\frown}{\phantom{A},\phantom{B}}}\underset{u_5}{\text{___}}\phantom{A},\quad\ldots$$
Let's find the terms of an arithmetic sequence starting from $u_1$:
$\qquad u_2 = u_1+d$
$\qquad u_3 = u_1+d+d$
$\qquad u_4 = u_1+d+d+d$
Follow this pattern and express $u_5$ and $u_6$ in terms of $u_1$ and $d$
$\qquad u_5 = $
$\qquad u_6 = $
To find $u_n$, how many $d$'s must be added to $u_1$?
For arithmetic sequences, $u_n=u_1+($
$)\times d$.
Example 2
Consider the sequence $2, 9, 16, 23, 30, …$
a) Find a formula for the general term $u_n$. Simplify your answer.
b) Find the 100th term of the sequence.
c) If $u_n=828$, find $n$.
d) Is $2341$ a term of the sequence?
a)
b)
c)
d)
Example 3
Find $k$ given that $3k+1, k$, and $-3$ are consecutive terms of an arithmetic sequence.
Example 4
Find the general term $u_n$ for an arithmetic sequence with $u_3=8$ and $u_8=-17$.
Example 5
Write down all six terms of an arithmetic sequence with $u_1=3$ and $u_6=12$. Separate the terms with commas.