We will investigate the manipulation of arithmatic sequences expressed as general nth terms.
An arithmatic sequences are expressed as following:
[tex]a_{1\text{ }},a_2,a_3,\ldots a_{n-1},a_n[/tex]Where,
[tex]\begin{gathered} a_1\colon FirstTerm_{} \\ a_n\colon\text{ nth Term} \\ n\colon\text{ Term number} \end{gathered}[/tex]An arithmatic sequence is defined by two parameters:
[tex]\begin{gathered} a_1\colon\text{ First Term} \\ d\colon\text{ common difference} \end{gathered}[/tex]Where,
[tex]d=a_n-a_{n-1}[/tex]The general formulation of the nth term in a arithmatic sequence is as follows:
[tex]a_n=a_1\text{ + ( n - 1 )}\cdot d[/tex]We are given the following arithmatic sequence terms:
[tex]a_3=11,a_{12}\text{ = }74[/tex]We are to determine the common difference ( d ) for the above sequence. We will use the general formulation to construct equations:
[tex]\begin{gathered} a_3=a_1\text{ + ( 3 - 1 )}\cdot d\text{ = }11 \\ a_{12}=a_1\text{ + ( 12 - 1 )}\cdot d\text{ = 74} \\ \\ a_1\text{ + 2}\cdot d\text{ = }11\ldots\text{ Eq1} \\ a_1\text{ +11}\cdot d\text{ = 74 }\ldots\text{ Eq2} \end{gathered}[/tex]We will solve the above two equations simultaneosuly:
[tex]\begin{gathered} -a_1\text{ - 2}\cdot d\text{ = -}11 \\ a_1\text{ +11}\cdot d\text{ = 74 } \\ ========== \\ 9d\text{ = 63} \\ d\text{ = 7 }\ldots\text{ Answer} \\ \end{gathered}[/tex]The common difference for the above sequence is:
[tex]7[/tex]