WHICH TERM OF THE SEQUENCE 20,77/4, 37/2,71/4.......IS THE FIRST NEGATIVE TERM?

Asked by  | 6th Feb, 2014, 12:24: PM

Expert Answer:

Dear Student,

20,77%2F4, 37%2F2, 71%2F4

2nd term - 1st term = 77%2F4 - 20 = 77%2F4 - 80%2F4 = -3%2F4.
3rd term - 2nd term = 37%2F2 - 77%2F4 = 74%2F4 - 77%2F4 = -3%2F4.
4th term - 3rd term = 71%2F4 - 37%2F2 = 71%2F4 - 74%2F4 = -3%2F4.
 
So the common difference is d = -3%2F4
Use the nth term formula for an arithmetic sequence.

an = a1 + (n-1)d

1st term = a1 = 20, d = -3%2F4

an = 20 + (n-1)%28-3%2F4%29

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

20 + (n-1)%28-3%2F4%29 < 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%262%2F3

So the 28th term is the first negative term.

That term is:

an = a1 + (n-1)d

a28 = 20 + (28-1)%28-3%2F4%29

a28 = 20 + (27)%28-3%2F4%29

a28 = 20 + %28-81%2F4%29

a28 = 80%2F4 - 81%2F4

a28 = -1%2F4


 

Answered by  | 6th Feb, 2014, 03:12: PM

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