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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  [#permalink]

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New post 19 Jun 2019, 15:56
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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) \(2^{20}\)
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Re: The sequence Xn is defined as follows: Xn = 2X(n-1) - 1  [#permalink]

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New post 19 Jun 2019, 17:36
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\(X_1\)=\(2^1+1\)
\(X_2\)= \(2* (2+1)-1\)= \(2^2+1\)
and so on
We can generalize any term of the sequence as
\(X_n\)=\(2^n+1\)
\(X_{20}\)=\(2^{20}+1\)
\(X_{19}\)=\(2^{19}+1\)

\(X_{20}\)-\(X_{19}\)=\([2^{20}+1]\)- \([2^{19}+1]\)
\(X_{20}\)-\(X_{19}\)= \(2^{20}-2^{19}\)=\(2^{19}\)


energetics wrote:
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) \(2^{20}\)
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The sequence Xn is defined as follows: Xn = 2X(n-1) - 1  [#permalink]

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New post 19 Jun 2019, 17:42
energetics wrote:
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) \(2^{20}\)


\(X_{1}=3\), \(X_{2}=5\), \(X_{3}=9\), \(X_{4}=17\), \(X_{5}=33\)

We can see that the \(n_{th}\) term of the sequence \(X_{n}\) \(=2^n + 1\)

Therefore \(X_{20}\)\(=2^{20}+1\) and \(X_{19}\)\(=2^{19}+1\)

\(X_{20}\)\(-\)\(X_{19}\)\(=2^{20}+1-2^{19}-1\)

\(=2^{19}(2-1)=2^{19}\)

Answer is (D)

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The sequence Xn is defined as follows: Xn = 2X(n-1) - 1   [#permalink] 19 Jun 2019, 17:42
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