Question 388661
<pre>
{{{a[7] = 33}}} and {{{a[22] = 138}}}

{{{a[n]=a[1]+(n-1)d}}}

Substitute n=7

{{{a[7]=a[1]+(7-1)d}}}
{{{a[7]=a[1]+6d}}}

Substitute 33 for {{{a[7]}}}

{{{33=a[1]+6d}}}

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{{{a[n]=a[1]+(n-1)d}}}

Substitute n=22

{{{a[22]=a[1]+(22-1)d}}}
{{{a[22]=a[1]+21d}}}

Substitute 138 for {{{a[22]}}}

{{{138=a[1]+21d}}}

So we have this system of equations:

{{{system(33=a[1]+6d, 138=a[1]+21d)}}}

Solve that by substitution and get {{{a[1]=-9}}} and {{{d=7}}}

So the rule is

{{{a[n]=a[1]+(n-1)d}}}

{{{a[n]=-9+(n-1)7}}}

{{{a[n]=-9+7(n-1)}}}

{{{a[n]=-9+7n-7}}}

{{{a[n]=7n-16}}}

To check substitute n = 7 and n = 22 and see if we get 33 and 138

{{{a[7]=7(7)-16}}}
{{{a[7]=49-16}}}
{{{a[7]=33}}}

{{{a[22]=7(22)-16}}}
{{{a[7]=154-16}}}
{{{a[7]=138}}}

It checks, so the answer is

{{{a[n]=7n-16}}}

Edwin</pre>