Question

Write a rule for the nth term of the geometric sequence given a_7=58, a_11=94

Answer 1

We are told the sequence is arithmetic. This means that the difference between one therm and the next is a constant.

We are also given two terms of the sequence. Let's see what their difference is

`a_(11)-a_7=94-58=36`

This means that, in general

`a_(k+4)-a_k=36`

With this, we can deduce that the difference between any two cnsecutive terms will be 9, for example

`a_7=58,a_6=49,a_5=40,a_4=31,a_3=22`

Indeed,

`a_7-a_3=58-22=36`

Now we should find the first term of the sequence, a₀, in order to find the rule for the nth term.

`a_2=13,a_1=4,a_0=-5`

In general, the rule for the nth term of an arithmetic sequence is given by

`a_n=a_0+d(n)`

where d is the difference between two consecutive term. In this case we have

`a_n=-5+9\cdot n`

with n=0,1,2,....