Question

How do I do this? Write a rule for the n-th term in the arithmetic sequence. a5=11 a11=29

Answer 1

The expression of the n-th term of an arithmetic sequence is

`a_n=a_1+(n-1)\cdot d`

The problem here is to find the initial value a1 and the value of d (common difference), then, let's write that equation for each value that we have, first, let's do n = 5

`\begin{gathered} a_5=a_1+(5-1)\cdot d \\ \\ 11=a_1+4d \end{gathered}`

Now let's repeat the process for n = 11

`\begin{gathered} a_(11)=a_1+(11-1)\cdot d \\ \\ 29=a_1+10d \end{gathered}`

Now we have a system of equations!

`\begin{cases}a_1+4d=11{} \\ a_1+10d=29{}\end{cases}`

To solve that system we can subtract the equations

`\begin{gathered} a_1-a_1+4d-10d=11-29 \\ \\ -6d=-18 \\ \\ d=(18)/(6) \\ \\ d=3 \end{gathered}`

We can find a1 now because we have the value of d, we can use any of the two equations we wrote to do it, I'll use the first one

`\begin{gathered} a_1+4d=11 \\ \\ a_1+4\cdot3=11 \\ \\ a_1+12=11 \\ \\ a_1=-1 \end{gathered}`

Now we have all the term to write the rule of the n-th term,

`\begin{gathered} a_n=a_1+(n-1)\cdot d \\ \\ a_n=-1+(n-1)\cdot3 \end{gathered}`

The rule for the n-th term is

`a_n=-1+3\cdot(n-1)`