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(26D) The difference method for sequences

Let’s look at an alternative way to find the general term of sequences.

Consider the sequence $u_n=1,4,9,16,25,…$
We can find the difference between two successive terms and make a new sequence.
$\Delta 1$, the first difference, is the sequence of differences between terms in the sequence $u_n$.
$\Delta 2$, the second difference, is the sequence of differences between terms in the sequence $\Delta 1$.

$$
\begin{array}{lccccccccc}
n & \textbf{1} & & \textbf{2} & & \textbf{3} & & \textbf{4} & & \textbf{5} \\ \hline
u_n & 1 & & 4 & & 9 & & 16 & & 25 \\ \hline
\Delta1 & & 3 & & 5 & & 7 & & 9 & \\ \hline
\Delta2 & & & 2 & & 2 & & 2 & & \\ \hline
\end{array}
$$

Linear Sequences

Consider the sequence $5,8,11,14,17,…$

Sequence
\begin{array}{lccccccccc}
n & \textbf{1} & & \textbf{2} & & \textbf{3} & & \textbf{4} & & \textbf{5} \\ \hline
u_n & 5 & & 8 & & 11 & & 14 & & 17 \\ \hline
\Delta1 & & & & & & & & & \\ \hline
\end{array}

a) List the first 4 terms of $\Delta1$ for the sequence.
Separate the terms by commas.
General Sequence $u_n=an+b$
\begin{array}{lccccccccc}
n & \textbf{1} & & \textbf{2} & & \textbf{3} & & \textbf{4} & & \textbf{5} \\ \hline
u_n & & & & & & & & & \\ \hline
\Delta1 & & & & & & & & & \\ \hline
\end{array}

We know that the general term for this sequence is $u_n=3n+2$
However, assume that we didn’t know this and let’s guess that the general term is going to be in the form $an+b$ where we have to find what $a$ and $b$ are. This is shown in the table on the right.

a) List the first 5 terms of the general sequence in terms of $a$ and $b$.
Separate the terms by commas and no spaces.
Sequence
\begin{array}{lccccccccc}
n & \textbf{1} & & \textbf{2} & & \textbf{3} & & \textbf{4} & & \textbf{5} \\ \hline
u_n & 5 & & 8 & & 11 & & 14 & & 17 \\ \hline
\Delta1 & & 3 & & 3 & & 3 & & 3 & \\ \hline
\end{array}
General Sequence $u_n=an+b$
\begin{array}{lccccccccc}
n & \textbf{1} & & \textbf{2} & & \textbf{3} & & \textbf{4} & & \textbf{5} \\ \hline
u_n & a+b & & 2a+b & & 3a+b & & 4a+b & & 5a+b \\ \hline
\Delta1 & & & & & & & & & \\ \hline
\end{array}

a) List the first 4 terms of $\Delta1$ for the general sequence.
Separate the terms by commas and no spaces.

b) Comparing the two tables, $a=$

c) Since $u_1=5$ for the sequence and $u_1=a+b$ for the general sequence, $b=$

d) Therefore, the general term $u_n=$

Quadratic Sequences

Sequence
\begin{array}{lccccccccc}
n & \textbf{1} & & \textbf{2} & & \textbf{3} & & \textbf{4} & & \textbf{5} \\ \hline
u_n & 3 & & 8 & & 15 & & 24 & & 35 \\ \hline
\Delta1 & & & & & & & & & \\ \hline
\Delta2 & & & & & & & & & \\ \hline
\end{array}
General Sequence $u_n=an^2+bn+c$
\begin{array}{lccccccccc}
n & \textbf{1} & & \textbf{2} & & \textbf{3} & & \textbf{4} & & \textbf{5} \\ \hline
u_n & & & & & & & & & \\ \hline
\Delta1 & & & & & & & & & \\ \hline
\Delta2 & & & & & & & & & \\ \hline
\end{array}

It is suggested that you copy these tables and write your answers down.

a) List the first 4 terms of $\Delta1$ for the sequence.
Separate the terms by commas and no spaces.

b) List the first 3 terms of $\Delta2$ for the sequence.
Separate the terms by commas and no spaces.

c) Since $\Delta2$ is constant for the sequence, let's guess that the general sequence will be in the form $u_n=an^2+bn+c$.
List the first 5 terms of the general sequence in terms of $a,b,c$.
Separate the terms by commas and no spaces.

d) List the first 4 terms of $\Delta1$ for the general sequence in terms of $a,b,c$.
Separate the terms by commas and no spaces.

e) List the first 3 terms of $\Delta2$ for the general sequence in terms of $a,b,c$.
Separate the terms by commas and no spaces.

f) Comparing the $\Delta2$ rows in the two tables, $2=2a$ so $a=$

g) Comparing the first term of $\Delta1$ in the two tables, $5=3a+b$ so $b=$

h) Comparing the first term of $u_n$ in the two tables, $3=a+b+c$ so $c=$

i) Therefore, the general term for this sequence is $u_n=$

General Term of Quadratic Sequences

Find the general term, $u_n$, for the following sequences.
*Write your answers in the form $an^2+bn+c$ where $a,b,c$ are integers.

a) $2,6,12,20,30,...$

b) $0,3,8,15,24,...$

c) $0,7,18,33,52,...$

d) $-1,-3,-9,-19,-33,...$

e) $2,0,0,2,6,...$

General Term of Other Sequences

[Challenging] Find the general term, $u_n$, for the following sequences.

a) $1,6,21,52,105,186...$

b) $-6,-4,10,42,98,184...$

Suggested Practice

(26D on P.543) #1,3-7