The first term is a = 1/4
The common ratio is d = 1/2.
The sum of geometric progression is S = 31/64.
The formula for the geometric progression is,
[tex]S=\frac{a(1-r^n)}{1-r}[/tex]Substitute the given values in the expression to obtain the value of n.
[tex]\begin{gathered} \frac{31}{64}=\frac{\frac{1}{4}(1-(\frac{1}{2})^n)}{1-\frac{1}{2}} \\ \frac{31}{64}=\frac{1}{2}(1-\frac{1}{2^n}) \\ \frac{1}{2^n}=1-\frac{31}{32} \\ 2^n=\frac{32}{32-31} \\ 2^n=2^5 \\ n=5 \end{gathered}[/tex]The nub er of terms is n = 5.
The last term of geometric progression is,
[tex]a_n=ar^{n-1}[/tex]Substitute the values in the equation to obtain the last term of series.
[tex]\begin{gathered} a_5=\frac{1}{4}(\frac{1}{2})^{5-1} \\ =\frac{1}{4}\cdot\frac{1}{16} \\ =\frac{1}{64} \end{gathered}[/tex]So last term of geometric progression is 1/64.
