Question

The $23 rd term in a certain geometric sequence is 16 and the $28th term in the sequence is 24. What is the $43 rd term?

Answer 1

81

Explanation:

Geometric sequence formula:
`a_n=ar^(n-1)`

Given:

• 
  `a_(23)=ar^(22)=16`
• 
  `a_(28)=ar^(27)=24`

Find common ratio (r):

`\begin{aligned}\implies (a_(28))/(a_(23)) =(ar^(27))/(ar^(22)) & =(24)/(16)\\ \implies r^5 & =\frac32\\ \implies r & = \sqrt[5]{\frac32} \end{aligned}`

Find initial term (a):

`\implies ar^(22)=16`

`\implies a(\sqrt[5]{\frac32} )^(22)=16`

`\implies a(\frac32} )^{(22)/(5)}=16`

`\implies a=\frac{16}{(\frac32)^{(22)/(5)}}`

Find the 43rd term:

`\implies a_(43)=ar^(42)`

`\implies a_(43)= \left(\frac{16}{(\frac32)^{(22)/(5)}}\right)(\sqrt[5]{\frac32} )^(42)`

`\implies a_(43)=16 \cdot (\frac32)^{-(22)/(5)} \cdot (\frac32)^{(42)/(5)}`

`\implies a_(43)=16 \cdot (\frac32)^4`

`\implies a_(43)=16 \cdot \left((3^4)/(2^4)\right)`

`\implies a_(43)=16 \cdot \left((81)/(16)\right)`

`\implies a_(43)=81`