help, succession formula

Answer: [tex]a_n=-3*\left ( -\frac{2}{3} \right )^n \ n>=0[/tex] Step-by-step explanation: Succession can be understood as a sorted collection of values that respond to a general term or rule. We need to find if these numbers are in arithmetic progression or geometric progression. In an arithmetic progression, every number is obtained as the previous number plus or minus a constant value called common difference. In a geometric progression, we get the next numbers as the previous one multiplied or divided by a constant value, called the common ratio.  If we try to find a possible common difference between first and second terms we get: [tex]2-(-3)=5[/tex] If it was an arithmetic progression, third term should be [tex]2+5=7[/tex] Which is obviously not true. Now let's try to find a possible common ratio by dividing second by first term [tex]r=\frac{2}{-3}=-\frac{2}{3}[/tex] Testing our value to find the third term we get [tex]a_3=2*(-\frac{2}{3})=-\frac{4}{3}[/tex] Since we have more terms to test: [tex]a_3=(-\frac{4}{3})*(-\frac{2}{3})=\frac{8}{9}[/tex] The given value is just as predicted The fourth term can be accurately predicted also: [tex]a_4=(\frac{8}{9})*(-\frac{2}{3})=\frac{16}{27}[/tex] Now we are sure it's a geometric progression, it can easily be stated the general term of the progression is [tex]a_n=a_1*r^n \ n>=0[/tex] [tex]a_n=-3*\left ( -\frac{2}{3} \right )^n \ n>=0[/tex]