In a G.P., the $3^{\text {rd }}$ term is 24 and the $6^{\text {th }}$ term is 192 .Find the $10^{\text {th }}$ term.

Step 1: Define the general term of a G.P.

The general term of a Geometric Progression (G.P.) is given by the formula $a_n = ar^{n-1}$, where $a_n$ is the $n^{\text{th}}$ term, $a$ is the first term, and $r$ is the common ratio.

Step 2: Formulate equations from given terms

We are given that the $3^{\text{rd}}$ term is 24 and the $6^{\text{th}}$ term is 192. Using the general term formula, we can write these as two equations: $ar^2 = 24$ and $ar^5 = 192$.

Step 3: Find the common ratio (r)

To find the common ratio $r$, we can divide the equation for the $6^{\text{th}}$ term by the equation for the $3^{\text{rd}}$ term. This simplifies to $r^3 = 8$, which means $r = 2$.

Step 4: Find the first term (a)

Now that we have the common ratio $r=2$, we can substitute it back into the equation for the $3^{\text{rd}}$ term ($ar^2 = 24$) to find the first term $a$. This gives us $a(2)^2 = 24$, which simplifies to $4a = 24$, so $a = 6$.

Step 5: Calculate the 10th term

Finally, we need to find the $10^{\text{th}}$ term. Using the general formula $a_n = ar^{n-1}$ with $a=6$, $r=2$, and $n=10$, we get $a_{10} = 6(2)^9$. Calculating $2^9 = 512$, we find $a_{10} = 6 \times 512 = 3072$.

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