For the harmonic sequences given below, how to find the indicated term?

(1) #1/10#, #1/6#, #1/2#,....; 9th term
(2) #1/15#, #1/13#, #1/11#,....; 9th term
(3) #1/4#,#1/14#, #1/24#,....; 6th term

1 Answer
Somebody N.
Jul 8, 2018

#(1)=color(blue)(-1/22)#, #(2)=color(blue)(-1)#, #(3)=color(blue)(1/54)#

Explanation:

A harmonic sequence is the reciprocal of an arithmetic sequence:

Notice the denominators in each sequence:

(1) #1/10,1/6,1/2#

#10,6,2#

Common difference:

#d=6-10=2-6=-4#

First term:

#a=10#

The nth term of an arithmetic sequence is given by:

#a+(n-1)d#

So for the 9th term we have:

#10+(9-1)(-4)=-22#

And for the harmonic sequence:

#-1/22#

We can do the same thing for 2 and 3:

(2)

#d=-2#

#a=15#

#n=9#

#15+(9-1)(-2)=-1#

#-1#

(3)

#d=10#

#a=4#

#n=6#

#4+(6-1)(10)=54#

#1/54#

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