Question 855053
Find t sub n if t sub 3 is equal to 15 and t sub 8 is equal to 50
<pre>
{{{t[3]=15}}}, {{{t[8]=50}}}

If the sequence is arithmetic, we use the formula:

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

three times. Once for n=3. once for n=8, and once using
the values for t<sub>1</sub> and d.

--------------------------

Substitute n = 3

{{{t[3]}}}{{{""=""}}}{{{t[1]}}}{{{""+""}}}{{{(3-1)d}}}

{{{t[3]}}}{{{""=""}}}{{{t[1]}}}{{{""+""}}}{{{(2)d}}}

Substitute 15 for t<sub>3</sub>

{{{15}}}{{{""=""}}}{{{t[1]}}}{{{""+""}}}{{{2d}}}

-------------------------

Substitute n = 8

{{{t[8]}}}{{{""=""}}}{{{t[1]}}}{{{""+""}}}{{{(8-1)d}}}

{{{t[8]}}}{{{""=""}}}{{{t[1]}}}{{{""+""}}}{{{(7)d}}}

Substitute 50 for t<sub>8</sub>

{{{50}}}{{{""=""}}}{{{t[1]}}}{{{""+""}}}{{{7d}}}

So you have this system of equations to solve:

{{{system(15=t[1]+2d, 50=t[1]+7d)}}}

Solve that system by substitution or elimination and
get

{{{t[1]=1}}} and {{{d=7}}}

---------------------------------

Therefore 

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

becomes

{{{t[n]}}}{{{""=""}}}{{{1}}}{{{""+""}}}{{{(n-1)7}}}

{{{t[n]}}}{{{""=""}}}{{{1+7(n-1)}}}

{{{t[n]}}}{{{""=""}}}{{{1+7n-7}}}

{{{t[n]}}}{{{""=""}}}{{{7n-6}}}

Edwin</pre>