Question #a84c7
1 Answer
Explanation:
The explicit rule for a geometric sequence is
In this problem, we know that
To do this, substitute these values into the equation and solve for
#a_n=a_1*r^(n-1)#
#36=a_1*(1/3)^(5-1)#
#36=a_1*(1/3)^(4)#
#36=a_1*(1/81)#
#a_1=36*81#
#color(blue)(a_1=2916)#
Now that we know
#a_n=a_1*r^(n-1)#
#a_n=2916*(1/3)^(n-1)#
Finally, the problem asked us to find the tenth term,
#a_n=2916*(1/3)^(n-1)#
#a_10=2916*(1/3)^(10-1)#
#a_10=2916*(1/3)^9#
#a_10=2916*(1/19683)#
#color(blue)(a_10=4/27)#
So, the tenth term is
You could also do this problem another way. Since you knew that
#a_5=36#
#a_6=1/3 * a_5 => 1/3 * 36 = 12#
#a_7=1/3 * a_6 => 1/3 * 12 = 4#
#a_8=1/3 * a_7 => 1/3 * 4 = 4/3#
#a_9=1/3 * a_8 => 1/3 * 4/3 = 4/9#
#a_10=1/3 * a_9 => 1/3 * 4/9 = color(blue)(4/27)#
This method may seem easier at first, but if the problem asked you find, say, the thirtieth term, it would be much easier to figure out the explicit rule and use that.