Question

Consider the explicit formulas for two sequences below.

Answer 1

Answer:

C. f(7) > g(10)

Step-by-step explanation:

We are given the explicit formulas as,

$f(n)=2^{n-1}-1$ and $g(n)=3n+6$

So, according to the options, we have,

A. f(5) < g(3)

So, $f(5)=2^{5-1}-1$ i.e. $f(5)=2^{4}-1$ i.e. $f(5)=16-1$ i.e. f(5) = 15.

And, $g(3)=3\times 3+6$ i.e. $g(3)=9+6$ i.e. g(3) = 15

Thus, f(5) = g(3).

It is not correct.

B. g(6) > f(6)

So, $f(6)=2^{6-1}-1$ i.e. $f(6)=2^{5}-1$ i.e. $f(6)=32-1$ i.e. f(6) = 31

And, $g(6)=3\times 6+6$ i.e. $g(3)=18+6$ i.e. g(3) = 24

Thus, g(6) < f(6)

It is not correct.

C. f(7) > g(10)

So, $f(7)=2^{7-1}-1$ i.e. $f(7)=2^{6}-1$ i.e. $f(7)=64-1$ i.e. f(7) = 63

And, $g(10)=3\times 10+6$ i.e. $g(10)=30+6$ i.e. g(10) = 36

Thus, f(7)>g(10)

It is correct.

D. g(8) = f(5)

So, $f(5)=2^{5-1}-1$ i.e. $f(5)=2^{4}-1$ i.e. $f(5)=16-1$ i.e. f(7) = 15

And, $g(8)=3\times 8+6$ i.e. $g(8)=24+6$ i.e. g(8) = 30

Thus, g(8) ≠ f(5)

It is not correct.

Thus, the only correct statement is C. f(7) > g(10).