Expert: Paul Klarreich - 12/12/2010

Question
I have a doubt in this question! I am trying to find the answer since a longwhile but am unable to get the correct answer according to the book.The Question is:
  Determine the explicit formula for the terms in the series: -2-5-8-11-14-17-20
     =a+(n-1)d
    =-2+(n-1)-3
    =-2+3n+3
    = 3n+1 (Answer)
But the book's answer is 7-3n!
I dont know how they came up with that answer?!Can you please help me?

Answer
Questioner:   Xiang
Country:  Canada
Category:  Advanced Math
Private:  No
 
Subject:  Sequences and Series
Question:  I have a doubt in this question! I am trying to find the answer since a long while but am unable to get the correct answer according to the book.The Question is:

Determine the explicit formula for the terms in the series: -2-5-8-11-14-17-20
    =a+(n-1)d
   =-2+(n-1)-3  << need ():  -2 + (n-1)(-3)
   =-2+3n+3     << wrong arithmetic:  -2 - 3n + 3
   = 3n+1 (Answer)  << should be  1 - 3n
But the book's answer is 7-3n!
I dont know how they came up with that answer?!Can you please help me?
................................
I think your sequence is meant to be:
-2,-5,-8,-11,-14,-17,-20,...

So this is a decreasing arithmetic sequence.

Now if  
A[0] = -2,
A[1] = -5, etc.

Then you should have

A[0] = a + 0d = -2
A[1] = a + 1d = -5

Then d = -3, and a = -2, so the sequence is given by:

A[n] = -2 - 3n

However, you can decide to shift the sequence by three terms:

Let  m = n + 3, and  n = m - 3

A[m] = -2 - 3(m - 3)

A[m] = -2 - 3m + 9
 
A[m] = 7 - 3m

So this will get your formula.  Why do it?  Don't ask.