Answer:
The additional words is [tex]2^n[/tex]
Explanation:
Given
[tex]B(n) = 2^n[/tex]
Required
Determine the additional words; i.e. [tex]B(n + 1) - B(n)[/tex]
From the given parameters, we have that;
B is a function of n
Such that;
[tex]B(n) = 2^n[/tex]
To calculate [tex]B(n+1)[/tex], we simply substitute n + 1 for n
[tex]B(n) = 2^n[/tex]
[tex]B(n + 1) = 2^{n + 1}[/tex]
Applying laws of indices
[tex]B(n + 1) = 2^{n} * 2^1[/tex]
[tex]B(n + 1) = 2^{n} * 2[/tex]
[tex]B(n + 1) = 2(2^{n})[/tex]
Calculating Additional Binary Code;
[tex]B(n + 1) - B(n)[/tex]
Substitute values for B(n + 1) and B(n)
[tex]B(n + 1) - B(n) = 2(2^n) - 2^n[/tex]
Express [tex]2^n[/tex] as [tex]2^ n * 1[/tex]
[tex]B(n + 1) - B(n) = 2(2^n) - 2^n * 1[/tex]
Express 1 as [tex]2^0[/tex]
[tex]B(n + 1) - B(n) = 2(2^n) - 2^n * 2^0[/tex]
Factorize
[tex]B(n + 1) - B(n) = 2^n(2 - 2^0)[/tex]
[tex]B(n + 1) - B(n) = 2^n(2 - 1)[/tex]
[tex]B(n + 1) - B(n) = 2^n(1)[/tex]
[tex]B(n + 1) - B(n) = 2^n[/tex]
Hence, the additional words is [tex]2^n[/tex]
