In a G.P., the 3rd term is 24 and the 6th term is 192 .Find the 10th term.
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Step-by-Step Solution
Step 1: Define the general term of a G.P.
The general term of a Geometric Progression (G.P.) is given by the formula an=arn−1, where an is the nth term, a is the first term, and r is the common ratio.
Step 2: Formulate equations from given terms
We are given that the 3rd term is 24 and the 6th term is 192. Using the general term formula, we can write these as two equations: ar2=24 and ar5=192.
Step 3: Find the common ratio (r)
To find the common ratio r, we can divide the equation for the 6th term by the equation for the 3rd term. This simplifies to r3=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 3rd term (ar2=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 10th term. Using the general formula an=arn−1 with a=6, r=2, and n=10, we get a10=6(2)9. Calculating 29=512, we find a10=6×512=3072.