Write a rule for the nth term of the arithmetic sequence a 10 = 2 , a 22 = - 8.8

Question

Write a rule for the nth term of the arithmetic sequence             a      10        =    2    ,          a      22        =    -    8.8

Cseszteq5j · Accepted Answer

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

∴ {\begin{cases} a_{1} + 9 d = 2 \\ a_{1} + 21 d = - 8.8 \end{cases}

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} + 21 d) - (a_{1} + 9 d) = - 10.8

12 d = - 10.8 \Rightarrow d = - 0.9

Now substitute d in any of the two equations to find

a_{1}

.

a_{1} - 8.1 = 2 \Rightarrow 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 - 0.9 n

Write a rule for the nth term of the arithmetic sequence a_10=2,a_22=−8.8