Question 1135564
{{{a[n] = a[1]*r^(n - 1 )}}}

if the second and fifth terms are {{{ a[2] =-21}}} and {{{ a[5] =567}}}, respectively

{{{-21 = a[1]*r^(2 - 1 )}}}

{{{-21 = a[1]*r}}}......solve for {{{a[1]}}}

{{{a[1]=-21/r}}}.......eq.1



{{{567 = a[1]*r^(5 - 1 )}}}

{{{567 = a[1]*r^4}}}......solve for {{{a[1]}}}

{{{a[1]=567/r^4}}}.......eq.2


from eq.1 and eq.2 we have


{{{567/r^4=-21/r}}}

{{{567/r^3=-21}}}

{{{r^3=567/-21}}}

{{{r^3=-27}}}

{{{r^3=-3^3}}}

{{{highlight(r=-3)}}}


now find first term

{{{a[1]=-21/-3}}}.......eq.1

{{{highlight(a[1]=7)}}}


so, your ormula is:

{{{a[n] =7*(-3)^(n-1)}}}