Question 695853
<pre>
20, {{{77/4}}}, {{{37/2}}}, {{{71/4}}}

2nd term - 1st term = {{{77/4}}} - 20 = {{{77/4}}} - {{{80/4}}} = {{{-3/4}}}.
3rd term - 2nd term = {{{37/2}}} - {{{77/4}}} = {{{74/4}}} - {{{77/4}}} = {{{-3/4}}}.
4th term - 3rd term = {{{71/4}}} - {{{37/2}}} = {{{71/4}}} - {{{74/4}}} = {{{-3/4}}}.
 
So the common difference is d = {{{-3/4}}}

Use the nth term formula for an arithmetic sequence.

a<sub>n</sub> = a<sub>1</sub> + (n-1)d

1st term = a<sub>1</sub> = 20, d = {{{-3/4}}}

a<sub>n</sub> = 20 + (n-1){{{(-3/4)}}}

We set that < 0 to make it negative, and solve for n:

20 + (n-1){{{(-3/4)}}} < 0

Multiply both sides by 4

80 + (n-1)(-3) < 0
80 + (-3)(n-1) < 0
   80 - 3(n-1) < 0
   80 - 3n + 3 < 0
       83 - 3n < 0
           -3n < -83
             n > {{{27&2/3}}}

So the 28th term is the first negative term.

That term is:

a<sub>n</sub> = a<sub>1</sub> + (n-1)d

a<sub>28</sub> = 20 + (28-1){{{(-3/4)}}}

a<sub>28</sub> = 20 + (27){{{(-3/4)}}}

a<sub>28</sub> = 20 + {{{(-81/4)}}}

a<sub>28</sub> = {{{80/4}}} - {{{81/4}}}

a<sub>28</sub> = {{{-1/4}}}

Edwin</pre>