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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## The first and second numbers in a sequence of numbers are tagged by: AAPL ##### This topic has 1 expert reply and 0 member replies ### Top Member ## The first and second numbers in a sequence of numbers are ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult Veritas Prep The first and second numbers in a sequence of numbers are plotted as the x and y coordinates, respectively, of a point on the coordinate plane, as are the third and fourth numbers, and all subsequent pairs in the sequence, and a line is formed by connecting these points, what would be the slope of the line? 1) Except for the first number, the sequence of numbers is formed by doubling the previous number and then subtracting 1. 2) The first number in the sequence is 3. OA A. ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1008 messages Followed by: 27 members Upvotes: 59 AAPL wrote: Veritas Prep The first and second numbers in a sequence of numbers are plotted as the x and y coordinates, respectively, of a point on the coordinate plane, as are the third and fourth numbers, and all subsequent pairs in the sequence, and a line is formed by connecting these points, what would be the slope of the line? 1) Except for the first number, the sequence of numbers is formed by doubling the previous number and then subtracting 1. 2) The first number in the sequence is 3. $$\left( {{a_1},{a_2}} \right)\,\,,\,\,\left( {{a_3},{a_4}} \right)\,\,,\,\,\left( {{a_5},{a_6}} \right)\,\,,\,\,\, \ldots \,\,\,\, \to \,\,\,{\text{all}}\,\,{\text{points}}\,\,{\text{belong}}\,\,{\text{to}}\,\,{\text{a}}\,\,{\text{certain}}\,\,{\text{line}}\,\,{\text{L}}$$ $${\text{?}}\,\,\,{\text{ = }}\,\,\,\,{\text{slop}}{{\text{e}}_{\,{\text{L}}}}\,\,\,\left( { = \,\,\frac{{{a_4} - {a_2}}}{{{a_3} - {a_1}}} = \,\frac{{{a_6} - {a_4}}}{{{a_5} - {a_3}}} = \ldots } \right)$$ $$\left( 1 \right)\,\,\,L\,\,\,:\,\,\,y = 2x - 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 2\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{SUFF}}.$$ $$\left( 2 \right)\,\,\,{a_1}\, = 3\,\,\,\,\, \Rightarrow \,\,\,{\text{trivial}}\,\,{\text{geometric}}\,\,{\text{bifurcation}}\,\,\,\, \Rightarrow \,\,\,\,\,{\text{INSUFF}}.$$ Alternate (algebraic) argument for statement (1): $$\left( 1 \right)\,\,{a_n} = 2 \cdot {a_{n - 1}} - 1\,\,\,,\,\,{\text{for}}\,\,{\text{all}}\,\,n \geqslant 2\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,? = \frac{{{a_4} - {a_2}}}{{{a_3} - {a_1}}} = \frac{{6\left( {{a_1} - 1} \right)}}{{\,3\left( {{a_1} - 1} \right)}}\, = 2\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.\,$$ $$\left( * \right)\,\,\,\left\{ \matrix{ {a_4} = 2 \cdot {a_3} - 1 = 2\left( {2 \cdot {a_2} - 1} \right) - 1 = 4 \cdot {a_2} - 3\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{a_4} - {a_2}\,\, = \,\,\,3\left( {{a_2} - 1} \right) = 3\left[ {\left( {2 \cdot {a_1} - 1} \right) - 1} \right] = 6\left( {{a_1} - 1} \right) \hfill \cr {a_3} = \,2 \cdot {a_2} - 1 = 2\left( {2 \cdot {a_1} - 1} \right) - 1 = 4 \cdot {a_1} - 3\,\,\,\, \Rightarrow \,\,\,\,\,\,{a_3} - {a_1}\,\, = \,\,\,3\left( {{a_1} - 1} \right) \hfill \cr} \right.$$ This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or https://GMATH.com.br (Portuguese version) Course release PROMO : finish our test drive till 30/Dec with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount! • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • Award-winning private GMAT tutoring Register now and save up to$200

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