Answer:
81
Step-by-step explanation:
Geometric sequence formula: [tex]a_n=ar^{n-1}[/tex]
Given:
Find common ratio (r):
[tex]\begin{aligned}\implies \dfrac{a_{28}}{a_{23}} =\dfrac{ar^{27}}{ar^{22}} & =\dfrac{24}{16}\\ \implies r^5 & =\dfrac32\\ \implies r & = \sqrt[5]{\frac32} \end{aligned}[/tex]
Find initial term (a):
[tex]\implies ar^{22}=16[/tex]
[tex]\implies a(\sqrt[5]{\frac32} )^{22}=16[/tex]
[tex]\implies a(\frac32} )^{\frac{22}{5}}=16[/tex]
[tex]\implies a=\dfrac{16}{(\frac32)^{\frac{22}{5}}}[/tex]
Find the 43rd term:
[tex]\implies a_{43}=ar^{42}[/tex]
[tex]\implies a_{43}= \left(\dfrac{16}{(\frac32)^{\frac{22}{5}}}\right)(\sqrt[5]{\frac32} )^{42}[/tex]
[tex]\implies a_{43}=16 \cdot (\frac32)^{-\frac{22}{5}} \cdot (\frac32)^{\frac{42}{5}}[/tex]
[tex]\implies a_{43}=16 \cdot (\frac32)^4[/tex]
[tex]\implies a_{43}=16 \cdot \left(\dfrac{3^4}{2^4}\right)[/tex]
[tex]\implies a_{43}=16 \cdot \left(\dfrac{81}{16}\right)[/tex]
[tex]\implies a_{43}=81[/tex]
