Which term of the sequence { 16/9, 4/3, 1, 3/4 ... } is 243/1024?

Dec 26, 2015

The $8$th

Explanation:

This is a geometric sequence with initial term $\frac{16}{9}$ and common ratio $\frac{3}{4}$.

The general term of the sequence can be written:

${a}_{n} = \frac{16}{9} \cdot {\left(\frac{3}{4}\right)}^{n - 1} = {\left(\frac{3}{4}\right)}^{n - 3}$ for $n = 1 , 2 , 3 , \ldots$

Observe that $243 = {3}^{5}$ and $1024 = {2}^{10} = {4}^{5}$.

Comparing this power with the expression $n - 3$, we find that:

${a}_{8} = \frac{243}{1024}$.