Question 322889
<pre><b>
If the 3rd and 7th term of G.P are 81 and 16 respectively.
Find the first 5th term of the G.P

{{{a[1]}}}{{{"="}}}{{{"__"}}}, 
{{{a[2]}}}{{{"="}}}{{{"__"}}},
{{{a[3]}}}{{{"="}}}{{{"81"}}},
{{{a[4]}}}{{{"="}}}{{{"__"}}},
{{{a[5]}}}{{{"="}}}{{{"__"}}},
{{{a[6]}}}{{{"="}}}{{{"__"}}},
{{{a[7]}}}{{{"="}}}{{{"16"}}}

Look at the subsequence of b's:

{{{b[1]}}}{{{""=""}}}{{{a[3]}}}{{{""=""}}}{{{"81"}}},
{{{b[2]}}}{{{""=""}}}{{{a[4]}}}{{{""=""}}}{{{"__"}}},
{{{b[3]}}}{{{""=""}}}{{{a[5]}}}{{{""=""}}}{{{"__"}}},
{{{b[4]}}}{{{""=""}}}{{{a[6]}}}{{{""=""}}}{{{"__"}}},
{{{b[5]}}}{{{""=""}}}{{{a[7]}}}{{{""=""}}}{{{"16"}}}

We use the formula:

{{{b[n]}}}{{{""=""}}}{{{b[1]*r^(n-1)}}}

{{{16}}}{{{""=""}}}{{{81*r^(5-1)}}}
{{{16}}}{{{""=""}}}{{{81*r^4}}}
{{{16/81}}}{{{""=""}}}{{{r^4}}}
{{{root(4,16/81)}}}{{{""=""}}}{{{root(4,r^4)}}}
{{{2/3}}}{{{""=""}}}{{{r}}}

The common ratio r for the subsequence is the same
as the common ratio for the original whole sequence.

So we multiply or divide by {{{r=2/3}}} as necessary to fill in
the blanks:

To find {{{a[2]}}} we divide {{{a[3]=81}}} by {{{r=2/3}}}:

{{{81}}}{{{"÷"}}}{{{2/3}}}{{{""=""}}}{{{81}}}{{{"x"}}}{{{3/2}}}{{{""=""}}}{{{243/2}}}

To find {{{a[1]}}} we divide {{{a[2]=243/2}}} by {{{r=2/3}}}:

{{{243/2}}}{{{"÷"}}}{{{2/3}}}{{{""=""}}}{{{243/2}}}{{{"x"}}}{{{3/2}}}{{{""=""}}}{{{729/4}}}

To find {{{a[4]}}} we multiply {{{a[3]=81}}} by {{{r=2/3}}}:

{{{81}}}{{{"x"}}}{{{2/3}}}{{{""=""}}}{{{54}}} 

To find {{{a[5]}}} we multiply {{{a[4]=54}}} by {{{r=2/3}}}:

{{{54}}}{{{"x"}}}{{{2/3}}}{{{""=""}}}{{{36}}} 

To check we'll go ahead and find {{{a[6]}}}

We can find it either by multiplying {{{a[5]}}} by {{{r=2/3}}} 
or dividing {{{a[7]}}} by {{{r=2/3}}}

If we find a[6] by multiplying {{{a[5]}}} by {{{r=2/3}}} 

{{{36}}}{{{"x"}}}{{{2/3}}}{{{""=""}}}{{{24}}} 

If we find a[6] by dividing {{{a[7]}}} by {{{r=2/3}}} 

{{{36}}}{{{"x"}}}{{{2/3}}}{{{""=""}}}{{{24}}} 


{{{16}}}{{{"÷"}}}{{{2/3}}}{{{""=""}}}{{{16}}}{{{"x"}}}{{{3/2}}}{{{""-""}}}{{{24}}}

So it checks.

Answers:

{{{a[1]}}}{{{"="}}}{{{"729/4"}}}, 
{{{a[2]}}}{{{"="}}}{{{"243/2"}}},
{{{a[3]}}}{{{"="}}}{{{"81"}}},
{{{a[4]}}}{{{"="}}}{{{"54"}}},
{{{a[5]}}}{{{"="}}}{{{"36"}}},
{{{a[6]}}}{{{"="}}}{{{"24"}}},
{{{a[7]}}}{{{"="}}}{{{"16"}}}

Edwin</pre>