Question 1108441
Teachers like formulas, but the problem is that the same formula could be written differently by different teachers.
With some luck, the formulas your teacher uses formulas that look like the ones I will type below.
{{{t[n]}}}{{{"="}}}{{{t[1]*r^((n-1))}}}
{{{S[n]}}}{{{"="}}}{{{t[1]*(r^n-1)/(r-1)}}}
The first formula, applied for {{{n=3}}} and {{{n=6}}} says that
{{{t[3]}}}{{{"="}}}{{{t[1]*r^2}}} and {{{t[6]}}}{{{"="}}}{{{t[1]*r^5}}} .
The problem gives us the value of those terms, so
{{{20=t[1]*r^2}}} and {{{160=t[1]*r^5}}} .
Dividing one by the other,
{{{160/20}}}{{{"="}}}{{{t[1]*r^5/(t[1]*r^2)}}}
{{{8=r^3}}}
{{{2^3=r^2}}}
{{{r=2}}}
Going back to {{{20=t[1]*r^2}}} and substituting {{{r=2}}} , we get
{{{20=t[1]*2^2}}}
{{{20=t[1]*4}}}
{{{20/4=t[1]}}}
{{{t[1]=5}}} .
Now we can apply the second formula for {{{n=5}}} ,
{{{S[5]}}}{{{"="}}}{{{t[1]*(r^5-1)/(r-1)}}} ,
and substituting the values found before for {{{r}}} and {{{t[1]}}} ,
{{{S[5]}}}{{{"="}}}{{{5*(2^5-1)/(2-1)}}}
{{{S[5]}}}{{{"="}}}{{{5*(32-1)/1}}}
{{{S[5]}}}{{{"="}}}{{{5*31}}}
{{{highlight(S[5]=155)}}}