The sequence Xn is defined as follows: Xn = 2X(n-1) - 1

Question

The sequence  X_{n}  is defined as follows:  X_{n} = 2X_{(n-1)}-1  whenever n is an integer greater than 1. If  X_1=3 , what is the value of  X_{20} - X_{19} ?

(A)  2^{16}

(B)  2^{17}

(C)  2^{18}

(D)  2^{19}

(E)  2^{20}

sanjitscorps18 · Answer

Xn=2X(n−1)−1 whenever n is an integer greater than 1.

X1=3,

Value of X20−X19 = 2X19 - 1 -X19 = X19 - 1

If we build the series (X1, X2, X3, X4 .....) we get 3, 5, 9, 17 ......

This is essentially 2^1 + 1 , 2^2 + 1 , 2^3 + 1 , 2^4 + 1 ......

Hence X19 = 2^19 + 1

=> X20−X19 = X19 - 1 = 2^19 + 1 - 1 = 2^19

IMO Option D

rayjaycleoful · Answer

Could someone explain how we get 5,9, 17... when we expand X2,X3,X4....?

Bunuel · Answer

The formula  X_{n} = 2X_{(n-1)} - 1  is used to determine the  n^{th}  term of the sequence  X_n , given the preceding  n-1^{th}  term,  X_{n-1} .

Starting with  X_1 = 3 , the subsequent term,  X_2 , is calculated as follows:

X_{2} = 2X_{1} - 1 = 2 * 3 - 1 = 5 .

Similarly:

X_{3} = 2X_{2} - 1 = 2 *5 - 1 = 9 .

And so on.

Check the links below for more:

12. Sequences 
     Theory   
 Sequences Made Easy - All in One Topic! 
 Math : Sequences & Progressions 
 Time to Tackle Geometric Progressions! 
     Questions   
 Special Numbers and Sequences Problems  
 DS Questions 
 PS Questions

For other subjects:
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Hope it helps.

shuvamp · Answer

Xn=2X(n−1)−1...... equation 1
X1=3

Solving equation 1 for n=2,3,4,5..... we get, 
X2=2X1-1=2x3-1=5
X3=9
X4=17
X5=33

we observe that,  difference between consecutive numbers in the sequence is in 2^n 
X2-X1=5-3=2
X3-X2=9-5=4=2^2
X4-X3=17-9=8=2^3
X5-X4=33-17=16=2^4

HENCE, X20-X19=2^19, OPTION D

luisdicampo · Answer

Deconstructing the Question  
Sequence definition:  X_n = 2X_{n-1} - 1  for  n > 1 .
Initial term:  X_1 = 3 .
Target: Value of  X_{20} - X_{19} .

Method 1: Pattern Recognition  
Let's compute the first few terms:
 X_1 = 3 
 X_2 = 2(3) - 1 = 5 
 X_3 = 2(5) - 1 = 9 
 X_4 = 2(9) - 1 = 17

Notice the relationship with powers of 2:
 X_1 = 2^1 + 1 
 X_2 = 2^2 + 1 
 X_3 = 2^3 + 1 
...
General Formula:  X_n = 2^n + 1 .

Now substitute into the target expression:
 X_{20} - X_{19} = (2^{20} + 1) - (2^{19} + 1) 
 = 2^{20} - 2^{19} 
Factor out  2^{19} :
 = 2^{19}(2 - 1) = 2^{19} .

Method 2: Algebraic Simplification  
Given  X_{20} = 2X_{19} - 1 .
Subtract  X_{19}  from both sides:
 X_{20} - X_{19} = 2X_{19} - 1 - X_{19} 
 X_{20} - X_{19} = X_{19} - 1 .

Using the pattern  X_{19} = 2^{19} + 1 :
 X_{20} - X_{19} = (2^{19} + 1) - 1 = 2^{19} .

Answer:  D

The sequence Xn is defined as follows: Xn = 2X(n-1) - 1

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The sequence Xn is defined as follows: Xn = 2X(n-1) - 1

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