• November 6th 2008, 09:48 AM
juliak
Key, just in case there are varying letters used
U - unknown number
n - number you're looking for
d - common difference

Find the general term for Un for an arithmetic sequence given that:
U4=37 and U10 =67

I know how to do it up to here, but I don't know what to do next:
U10-U4=6d
--> U10-U4=67-37=30
6d=30
d=5

So I have the common difference but I don't know what to do with it now, even if I just insert it into the equation, I don't have the other values...

Un=U1+(n-1)d
Un=U1+(n-1)(5)
• November 6th 2008, 10:58 AM
Soroban
Hello, juliak!

Quote:

Find the general term $U_n$ for an arithmetic sequence given that:
. . . $U_4=37\:\text{ and }\:U_{10} =67$

We're expected to know that the $n^{th}$ term is: . $U_n \:=\:U_1 + (n-1)d$

. . where: . $\begin{Bmatrix}U_1 &=& \text{First term} \\ n &=& \text{no. of terms} \\ d &=& \text{common diff.} \end{Bmatrix}$

We're told that: . $\begin{array}{cccc}U_4 = 37 & \Rightarrow & U_1 + 3d \:=\:37 & {\color{blue}[1]} \\
U_{10} = 67 & \Rightarrow & U_1 + 9d \:=\:67 & {\color{blue}[2]}\end{array}$

Now solve the system of equations . . .

. . Subtract [1] from [2]: . $6d \:=\:30 \quad\Rightarrow\quad \boxed{d \:=\:5}$

. . Substitute into [1]: . $U_1 + 3(5) \:=\:37 \quad\Rightarrow\quad \boxed{U_1 \:=\:22}$

The sequence is: . $22,\:27,\:32,\:37,\:42,\:27,\:\hdots$

The general term is: . $U_n \:=\:22 + (n-1)5 \quad\Rightarrow\quad\boxed{ {\color{blue}U_n \:=\:5n + 17}}$