Question #a84c7

1 Answer
Fleur
Jul 31, 2017

#a_10=4/27#

Explanation:

The explicit rule for a geometric sequence is #a_n=a_1*r^(n-1)#, where #a_n# is the #n^(th)# term, #a_1# is the first term, and #r# is the common ratio.

In this problem, we know that #r=1/3#, and when #n=5#, #a_5=36#. However, we still have to figure out #a_1# to write the explicit rule for this particular sequence.

To do this, substitute these values into the equation and solve for #a_1#.

#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_1#, we have all the components to write the explicit rule for this sequence.

#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_10#.

#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 #4/27#.

You could also do this problem another way. Since you knew that #a_5 = 36# and #r=1/3#, you could keep multiplying each previous term by #1/3# until you reached #a_10#.

#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.

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