The 3rd and 6th terms of a geometric sequence are -75 and -9375 respectively. Find the first term, the common ratio and the explicit rule for the sequence

Answer 1

So we know the geometric sequence is a_n = a*r ^n-1

third term implies a₃ = a * r^3-1 this is -75 = a * r²

sixth term implies a₆ = a * r^6-1 this is -9375 = a * r⁵

From the first equation a₃ get a to be alone so a = -75/r² will plug this into the second equation to find r

-9375 = (-75/r²) * r⁵ this gives us 125 = r³ implies r = 5 plug this into the first equation to find a

-75 = a * (5)² this is -75 = a * 25 so a = -3

So we have a_n = -3 * 5^n-1

Hope this helps if you have any questions let me know.