Question 764814


If there is such a common ratio {{{r}}}, then this is a {{{geometric}}} sequence. 

If this is a geometric series, construct the formula:
 
{{{f(n) = f(1) * r^(n-1)}}}

use the formula to find {{{r}}} in our example:

{{{n}}}| {{{1}}}, {{{2}}}, {{{3}}}, {{{4}}}
{{{f(n)}}}| {{{4}}}, {{{20}}}, {{{100}}} ,{{{500 }}}

{{{f(1) = 4}}}.....plug in {{{f(n) = f(1) * r^(n-1)}}}

{{{f(n) = 4* r^(n-1)}}}

{{{f(2) = 20}}} 

{{{f(2) = 4* r^(2-1)=20}}}

=> {{{4* r^(2-1)=20}}}...solve for {{{r}}}

 {{{r^1=20/4}}}

 {{{r=5}}}

then sequence formula is: {{{f(n) = 4* 5^(n-1)}}}

check it:

{{{f(n) = 4* 5^(n-1)}}}

{{{n}}}| {{{1}}}, {{{2}}}, {{{3}}}, {{{4}}}
{{{f(n)}}}| {{{4}}}, {{{20}}}, {{{100}}} ,{{{500 }}}

{{{f(1) = 4* 5^(1-1)}}}
{{{f(1) = 4* 5^0}}}
{{{f(1) = 4* 1}}}
{{{f(1) = 4}}}..............true

{{{f(2) = 4* 5^(2-1)}}}
{{{f(2) = 4* 5^1}}}
{{{f(2) = 4* 5}}}
{{{f(2) = 20}}}..............true

{{{f(3) = 4* 5^(3-1)}}}
{{{f(3) = 4* 5^2}}}
{{{f(3) = 4* 25}}}
{{{f(3) = 100}}}..............true

{{{f(4) = 4* 5^(4-1)}}}
{{{f(4) = 4* 5^3}}}
{{{f(4) = 4* 125}}}
{{{f(4) = 500}}}..............true

since all true, your answer is: B. {{{f}}} is a geometric sequence with {{{r =5}}}