Arithmetic help plz...?

Question

If the third and fourth terms of an arithmetic progression are increased by 2 and 7 respectively, then the first four terms form a geometric progression. Which term of the arithmetic progression is 2008?

How do you do this?

Answer 1

Term number = 501

I'm so proud of myself! I actually figured it out!

Let t = the first term of the sequence.
Let d = the constant difference in the arithmetic sequence.

Therefore, the arithmetic sequence is:
t, t+d, t+2d, t+3d
As you can see, t+2d is the third term and t+3d is the fourth term.

Also, the geometric sequence is
t, t+d, t+2d+2, t+3d+7

--------------------------------
-------------Section2------------------
By definition, the consecutive terms of the geometric sequence have a constant ratio. So,

(t+d) / t =
(t+2d+2) / (t+d)

"The 2nd term / the 1st term = the 3rd term / the 2nd term"
It will help if you write it out on paper, it looks better and it will be easier to see what I'm doing. Now cross-multiply:

t(t+2d+2) = (t+d)^2
t^2 + 2dt + 2t = t^2 + 2dt + d^2

Simplify:
2t = d^2

-----------Section 3-------------------------
-----------------------------------------
Here is the geometric sequence again (from above):
t, t+d, t+2d+2, t+3d+7
Now we do the same thing as in Section 2 but like this:

(t+d) / t =
(t+3d+7) / (t+2d+2)

Follow?
"The 2nd term / the 1st term = the 4th term / 3rd term"

Cross multiply:
t(t+3d+7) = (t+d)(t+2d+2)

t^2 + 3dt + 7t =
t^2 + 2dt + 2t + dt + 2d^2 + 2d

t^2 + 3dt + 7t =
t^2 + 3dt + 2t + 2d^2 + 2d

Simplify:
5t = 2d^2 + 2d

----------Section 4------------------
---------------------------------------

Our two results from Sections 2 and 3 are:

Equation A: 2t = d^2
Equation B: 5t = 2d^2 + 2d

Multiply all terms of Equation A by 5/2 to get:

Equation C: 5t = (5/2)d^2

Subtract Equation C from Equation B to get
0 = 2d - (d^2)/2
0 = 4d - d^2

Now, multiply all terms by -1

0 = d^2 - 4d
0 = d(d+4)
d = 0 or 4
d = 0 makes no sense, so d must equal 4.
By Equation A, 2t = 16
t=8

d = 4
t = 8

--------------Section 5---------
-----------------------------------------
So, the arithmetic sequence is
8, 12, 16, 20
And the geometic sequence is therefore
8, 12, 18, 27
(r=3/2) But that doesn't really matter.

Set up your equation for 2008:

2008 = 8 + 4(n-1)
where n is the term number.

2008 = 8 + 4n - 4
2004 = 4n
n = 501