Question 935611
the formula for the exponential sequence is:


An = A0 * B^n


An is the nth term in the sequence.
A0 is the first term in the sequence
B is the base that is being raised to the exponent.
B is the common ratio.
n is the number of the term in the sequence.
n is also the exponent that B is being raised to.


for the second term in the sequence, you get:


A2 = A0 * B^2


since A2 is equal to 9, the equation becomes:


9 = A0 * B^2


for the fourth term in the sequence, you get:


A4 = A0 * B^4


since A4 is equal to 81, the equation becomes:


81 = A0 * B^4


your 2 equations are:


9 = A0 * B^2
81 = A0 * B^4


solve each of these equations for A0 and you get:


A0 = 9 / B^2
A0 = 81 / B^4


by substituting 9 / B^2 for A0 from the first equation, your second equation  of A0 = 81 / B^4 becomes:


9 / B^2 = 81 / B^4


multiply both sides of this equation by B^4 and divide both sides of this equation by 9 and you get:


B^4 / B^2 = 81 / 9


simplify to get:


B^2 = 9


take the square root of both sides of this equation and you get:


B = plus or minus 3.


B can't be negative, so B has to be equal to 3.


that's your solution.


the common ratio, which is B, is equal to 3.


you can also solve for A1 by replacing B with 3.


from the above equations for A0, you have:


A0 = 9 / B^2
A0 = 81 / B^4


replace B with 3 and those equations becomes:


A0 = 9 / 3^2 = 9 / 9 = 1
A0 = 81 / 3^4 = 81 / 81 = 1


A1 is equal to 1.
B is equal to 3.


An is equal to 1 * 3^n


when n is equal to 0, An is equal to 1 * 3^0 which is equal to 1
when n is equal to 1, An is equal to 1 * 3^1 which is equal to 3
when n is equal to 2, An is equal to 1 * 3^2 which is equal to 9
when n is equal to 3, An is equal to 1 * 3^3 which is equal to 27
when n is equal to 4, An is equal to 1 * 3^4 which is equal to 81