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(A) The given geometric progression is $5, - \frac{5}{2}, \frac{5}{4}, - \frac{5}{8}, \dots$Here,the first term $a = 5$ and the common ratio $r = \fra

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$11$

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(A) The given geometric progression is $5, - \frac{5}{2}, \frac{5}{4}, - \frac{5}{8}, \dots$Here,the first term $a = 5$ and the common ratio $r = \frac{-5/2}{5} = -\frac{1}{2}$.The $n^{th}$ term of a geometric progression is given by $T_n = ar^{n-1}$.Given $T_n = \frac{5}{1024}$,we have:$\frac{5}{1024} = 5 \left( -\frac{1}{2} \right)^{n-1}$Dividing both sides by $5$,we get:$\frac{1}{1024} = \left( -\frac{1}{2} \right)^{n-1}$Since $1024 = 2^{10}$,we can write $\frac{1}{1024} = \left( -\frac{1}{2} \right)^{10}$ because $(-1)^{10} = 1$.Thus,$\left( -\frac{1}{2} \right)^{10} = \left( -\frac{1}{2} \right)^{n-1}$.Comparing the exponents,we get $10 = n - 1$,which implies $n = 11$.

If the $n^{th}$ term of the geometric progression $5, - \frac{5}{2}, \frac{5}{4}, - \frac{5}{8}, \dots$ is $\frac{5}{1024}$,then the value of $n$ is:

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