What is the coefficient of determination for this data set?
Years after 2000
- 20-12-9-4 6 8 9 12
Federal minimum wage ($)3. 103. 353. 804. 255. 155. 155. 856. 557. 25
Data form U.S. Department of Labor 2012
Ο
O 02
Ο
Ο 0.91
Ο
0.95
Ο

1st,since this GP is convergent that means the common ratio r <1
2nd, sum of a GP = a₁(1-rⁿ)/(1-r), where a₁ = 1st term and n=number of terms
3rd, for any convergent GP, r<1 and the sum of all terms =a₁/(1-r): Why?
[since r<1 → lim rⁿ when n→∞, =0 in the formula of the 2nd)]
Now let's  solve :

a) Sum = a₁(1-r³)/(1-r) = 19 (sum of the first 3 terms)
b) Σ(Sum) = a₁/(1-r) = 27 (sum of all terms of this CONVERGENT GP)

Divide a) by b):

[a₁(1-r³)/(1-r)] / [a₁/(1-r)] = 19 /27 ↔ [a₁(1-r³)/(1-r)] x [(1-r)/a₁]=19/27.
Simplify:
(1-r³) = 19/27
-r³ = 19/27 - 1
r³ = 8/27

r = ∛(8/27)

r = 2/3 and a₁ = 9 (Plug r in the Σ sum)
Hence first term a₁ = 9
and common ration r =2/3