Question

In a G.P. the 3rd term is 24 and 6th term is 192. Find the common ratio​

Answer 1

✴Question:-

In a G.P. the 3rd term is 24 and 6th term is 192. Find the common ratio.

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✴Solution:-

→Given:-

• 3rd term=24
• 3rd term=246th term=192

→To Find:-

• Common Ration.

→Concept:-

• Simple G.P Problem.

→Formula:-

• $a_{n} = {ar}^{n - 1}$

→Calculation:-

We Know that,

$a_{n} = {ar}^{n - 1}$

$Where\:a_{n} = {n}^{th} \: term \: of \: G.P$

• n is the number of terms.
• a is the first term.
• r is the common ratio.

Here,

3rd term is 24.

$i.e. \: a_{3}=24$

$Putting \: \: a_{n}=24,n=3\:in\:a_{n}\:Formula$

$24 = {ar}^{3 - 1}$

$24 = {ar}^{2}$

${ar}^{2} = 24 \: \: \: \: \: \: \: ...(1)$

Similarly,

Given 6th term is 192

$i.e. \: a_{6} =192$

$Putting\:a_{n}=192,n=6\:in\:a_{n} \: Formula$

$192 = {ar}^{6 - 1}$

$195 = {ar}^{5}$

${ar}^{5} = 192 \: \: \: \: \: \: \: ...(2)$

Now Our equation are

${ar}^{2} = 24 \: \: \: \: \: ...(1)$

$and ,\: {ar}^{5} = 192 \: \: \: \: \: ...(2)$

Dividing (2) by (1)

$\frac{ {ar}^{5} }{ {ar}^{2} } = \frac{192}{24}$

$\frac{ {r}^{5} }{ {r}^{2} } = 8$

${r}^{5 - 2} = 8$

${r}^{3} = 8$

${r}^{3} = 2 \times 2 \times 2$

${r}^{3} = { (2) }^{3}$

$r = 2$

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✴Final Answer:-

So,r = 2

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