A sequence p1, p2, p3, ... pn is such that p1 = 3 and pn+1 = 2pn − 1

Question

A sequence  p_1 ,  p_2 ,  p_3 , ...  p_n  is such that  p_1 = 3  and  p_{n+1} = 2p_n − 1  for n ≥ 1, then  p_{10} − p_9 =

(A)  2^9

(B)  2^{10} − 1

(C)  2^{10}

(D)  2^{11} − 1

(E)  2^{11}

vanam52923 · Answer

So p2=2p1-1=2*3-1=5

p3=2p2-1=2*5-1=9

so series will be like

3,5,9,17,33.....

so 5-3=2=2^1
9-5=4=2^2
17-9=8=2^3

so 2nd term -1st term=2^1
3rd term-2nd term=2^2
like wise 
10th term-9th term=2^9

so A is answer

chetan2u · Answer

choices should tell you that the answer has to be in terms of   2^n . 
 keeping this in mind we can work forward..

p_{n+1} = 2p_n − 1...........p_{n+1} = p_n+p_n − 1.............p_{n+1} -p_n=p_n − 1  so  p_{10} − p_9 =p_9-1 
so let us find  p_9

now  p_1=3=2^1+1 ..
 p_2=2*p_1-1=2*3-1=5=2^2+1 ..
 p_3=2*p_2-1=2*5-1=9=2^3+1 
thus we get term for nth term
 p_n=2^n+1  and  p_9=2^9+1

and  p_{10} − p_9 =p_9-1=2^9-1+1=2^9

A

GMATinsight · Answer

To answer such a question one should always write a few terms (as many as you can calculate till 4th or 5th step) and then follow the patterns that question seeks at 10th term

p_1 = 3 
 p_2 = 2*3-1 = 5   i.e.  p_2 − p_1 =5-3 = 2^1  
 p_3 = 2*5- 1 = 9   i.e.  p_3 − p_2 =9-5 = 2^2  
 p_4 = 2*8- 1 = 17   i.e.  p_4 − p_3 =17-9 = 2^3  
 p_5 = 2*17- 1 = 33   i.e.  p_5 − p_4 =33-17 = 2^4  
.....
.....
 .....  Following the same pattern we can deduce that  p_{10} − p_9 = 2^9

jfranciscocuencag · Answer

Hello everyone!

How do we know that Pn equals P1?

Kind regards!

jfranciscocuencag · Answer

Hello everyone!

Could someone please explain to me how to pick up the right value?

p_{n+1} = 2p_n − 1  for n ≥ 1

How to be sure to which value they are referring with :

p_{n+1} = 2p_n − 1

P1  =  3 
P2 = 2(3) - 1 = 5

But, I got confused then I had to pick up the value of Pn, is Pn the result of the whole term or is just the number of the term?

Pn can be either  3  or  1 ?

Can someone help me please?

NinetyFour · Answer

Hi, is there a trick to know what equation to find the pattern? I would not have immediately tried to subtract the term with the proceeding term, and I do not think I have the time to brainstorm on the test to find a pattern like you described above.

Paras96 · Answer

given p1=3
therefore p2=2p1-1=5
similarly p3=9
p4=17

now consider, p2-p1=2=2^1
p3-p2=4=2^2
p4-p3=8=2^3
.
.
.
pn+1-pn=2^n

Hence p10-p9=2^9
Hence A

bumpbot · Answer

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A sequence p1, p2, p3, ... pn is such that p1 = 3 and pn+1 = 2pn − 1

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A sequence p1, p2, p3, ... pn is such that p1 = 3 and pn+1 = 2pn − 1

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