Question

The seventh term, u7 , of a geometric sequence is 108. the eighth term, u8 , of the sequence is 36 . (a) write down the common ratio of the sequence. [1 mark] (b) find u1 . [2 marks] the sum of the first k terms in the sequence is 118 096 . (c) find the value of k . [

Answer 1

To solve the first part we are going to use the formula for the nth therm of geometric sequence:
`a_(n)=ar^(n-1)`
where

`a_(n)` is the nth term

`a_(1)` is the first term

`r` is the ratio

`n` is the position of the term in the sequence

a. The ratio of a geometric sequence is
`r= (a_(n))/(a_(n-1))`. We know for our problem that
`a_n=u_(8)=36` and
`a_(n-1)=u_(7)=108`. Lets replace those values in our formula to find
`r`:

`r= (36)/(108)`

`r= (1)/(3)`

We can conclude that the ratio of our geometric sequence is
`r= (1)/(3)`.

b. To find
`a_(1)` we are going to use the formula for the nth therm of geometric sequence, the ratio, and the given fact that
`u_(7)=108`:

`a_(n)=ar^(n-1)`

`108=a_(1)( (1)/(3))^(7-1)`

`108=a_(1)( (1)/(3))^(6)`

`108=a_(1)( (1)/(729) )`

`a_(1)= (108)/( (1)/(729) )`

`a_(1)=78732`

We can conclude that the first therm,
`a_(1)`, of our geometric sequence is 78732.

c. To solve this one we are going to use the formula for the sum of the first nth terms of a geometric sequence:
`S_(k)=a_(1)( (1-r ^k))/(1-r) )`
where

`S_(k)` is the sum of the first
`k` terms

`a_(1)` is the first term

`r` is the common ratio

`k` is the number of terms

We know for our problem that
`S_(k)=118096`, and we also know for previous calculations that
`a_(1)=78732` and
`r= (1)/(3)`. So lets replace those values in our formula to find
`k`:

`S_(k)=a_(1)( (1-r ^k))/(1-r) )`

`118096=78732[ (1-( (1)/(3))^k )/(1- (1)/(3) ) ]`

`(118096)/(78732) = (1-( (1)/(3))^k )/( (2)/(3) )`

`k=10`

We can conclude the the sum of the first 10 terms of our geometric sequence is 118096.