Find a formula for
the arithmetic sequence a_{n} given that a_{5}=54 and a_{10}=89.

A_{n}, the
arithmetic sequence always has the form a_{n}=a_{1}+d(n-1)
where d is the common difference and a_{1} is the first term.

Plug in the information given to us in the problem to form a system of equations

a_{5}=54=a_{1}+d(5-1)=a_{1}+4d

54=a_{1}+4d

a_{10}=89=a_{1}+d(10-1)=a_{1}+9d

89=a_{1}+9d

Subtract the two equations to obtain

5d=35

d=7

By back
substituting d=7 into either of the two equations, we can find a_{1}

a_{1}+4(7)=54

a_{1}=54-28=26

a_{n}=a_{1}+d(n-1)

a_{n}=26+7(n-1)=26+7n-7=19+7n