Given the sequence 162,54, 18,6 ...find s7

Question

help please

Answer 1

With most sequences at a high school level, you're being asked what basic operation (+/-/*/÷) is being used at each step, and that step is usually repeated.

 

In this case, we look at what the sequence is originally doing:

 

162, 54, 18, 6...

 

It's not subtraction, as the sequence is not the same each time (by extension, it is also not addition). So we see that the sequence is being divided by a term each time (also known as a geometric sequence). We establish this common ratio by dividing two sequential terms (terms that are next to one another).

 

I choose 18 and 6 in this case, as they are relatively small numbers. 6/18 reduces to 1/3 which is our common ratio.

 

With that in mind, I know my first time a₁ = 162. I plug in each of these terms into the generic formula for a geometric sequence:

 

a_n = a₁(r)^(n-1)

 

Where a_n is our nth term, a₁ is our first term and r is our common ratio.

 

a_n = 162(1/3)^(n-1)

 

We then make the replacement for n to find our nth term, n being in this case 7 (7th term, nth term, see?)

 

a₇ = 162(1/3)^(7-1)

 

a₇ = 162(1/729)

 

a₇ = 2/9

Answer 2

This is a geometric sequence.  You can find the common ratio (r) by dividing any term by its preceding term:

 

     r = 54/162 = 18/54 = 6/18 = 1/3

 

The explicit formula is:

 

     a_n = a₁*r^n-1

     a_n = 162*(1/3)^n-1

 

where a₁ = 162 and r = 1/3.  For the seventh term (n=7):

 

    a₇ = 162*(1/3)^7-1

    a₇ = 162*(1/3)⁶

    a₇ = 162*(1/729) = 2*3⁴/3⁶ = 2/9