Question 493683
the third and sixth terms of a geometric sequence are-75 and -9375 respectively. Find the first term, common ratio, and an explict rule for the nth term.
======================================================
The general formula for the n-th term of a geometric sequence is:
{{{a_n = a*r^(n-1)}}}
where a = the 1st term, r = the common ratio
Given: a_3 = -75, a_6 = -9375
So we have two equations:
{{{a_3 = -75 = a*r^2}}}
{{{a_6 = -9375 = a*r^5}}}
From the 1st equation we have
{{{a = -75/r^2}}}
Substituting this into the equation for a_6 gives
{{{-9375 = (-75/r^2)*r^5}}}
{{{r^3 = 125}}}
This gives r = 5
Use the formula for a_3 above to solve for a:
{{{-75 = a*5^2}}}
This gives a = -3
So the rule for the nth term is:
{{{a_n = -3*5^(n-1)}}}