How do you write a rule for the nth term of the arithmetic sequence #a_10=2, a_22=-8.8#?

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Answer 1 · 2018-07-14T17:35:36

#a_n=11 - 0.9n#

The #n#-th term of an arihmetic progression is defined as:

#a_n = a_1 + (n-1)d#

where #d# is the common difference.

We know that

#a_10 = 2#
#a_22 = -8.8#

#:. {(a_1+9d=2),(a_1+21d = -8.8) :}#

To find the general rule, we have to solve this system of linear equation. It is fairly easy; substract the first equation from the second:

#(a_1+21d)-(a_1+9d) = -10.8#

#12d = -10.8 => d = -0.9#

Now substitute #d# in any of the two equations to find #a_1#.

#a_1 - 8.1 = 2 => a_1 = 10.1#

Hence the general rule of the #n#-term is going to be

#a_n = 10.1-0.9(n-1)#

Which is equivalent to

#a_n =11 - 0.9n#