itomath.com

Arithmetic Sequences

a) Is the sequence $3,6,9,12,15,...$ arithmetic, geometric, or neither?

b) Find a formula for the general term of $3,6,9,12,15,...\qquad u_n=$

c) Find a formula for the general term of $4,7,10,13,16,...\qquad u_n=$

d) Find a formula for the general term of $5,8,11,14,17,...\qquad u_n=$

e) Find a formula for the general term of $6,9,12,15,18,...\qquad u_n=$

f) Find a formula for the general term of $10,13,16,19,22,...\qquad u_n=$

g) Find a formula for the general term of $100,103,106,109,112,...\qquad u_n=$

h) Find a formula for the general term of $2,5,8,11,14...\qquad u_n=$

i) Find a formula for the general term of $1,4,7,10,13,...\qquad u_n=$

j) Find a formula for the general term of $0,3,6,9,12,...\qquad u_n=$

k) Find a formula for the general term of $-1,2,5,8,11,...\qquad u_n=$

l) Find a formula for the general term of $-100,-97,-94,-91,-88,...\qquad u_n=$

General Term of Arithmetic Sequences

Find the general term, $u_n$, for the following sequences.
*Write your answers in the form $an+b$ where $a$ and $b$ are numbers.
Also, find the 100th term, $u_{100}$

a) $1,3,5,7,9,\ldots \qquad u_n=$

b) $u_{100}=$

c) $15,18,21,24,27,\ldots \qquad u_n=$

d) $u_{100}=$

e) $6,17,28,39,50,\ldots \qquad u_n=$

f) $u_{100}=$

g) $-11,-14,-17,-20,-23,\ldots \qquad u_n=$

h) $u_{100}=$

i) $13,5,-3,-11,-19,\ldots \qquad u_n=$

j) $u_{100}=$

k) *Write your answer in the form $\frac{\boxed{\phantom{1}}}{\boxed{\phantom{3}}}n+\frac{\boxed{\phantom{1}}}{\boxed{\phantom{6}}}:\qquad$ $\frac{1}{2},\frac{5}{6},\frac{7}{6},\frac{3}{2},\frac{11}{6},\ldots \qquad u_n=$

l) $u_{100}=$

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.

Exercises

(Core 5B.1 on P.94) #2, 5, 6, 9, 10b, 13, 15, 19