Question

If in a G.P. t5: t3 = 7:9, then t9:t5​

Answer 1

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❚ QuEstiOn ❚

If in a G.P. $\ \ {t_5: t_3 = 7:9}$ , then $\ \ {t_9: t_5 }$ = ?

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❚ ANsWeR ❚

✺ Given :

• $\ \ {t_5: t_3 = 7:9}$

$\longrightarrow\ \ {\dfrac{t_5}{ t_3} =\dfrac{7}{9}}$

✺ To FinD :

• $\ \ {t_9: t_5 }$ = ?

✺ Explanation :

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❏ G.P. Series:-

If in a G.P. series "a" be the first term and "r" be the common ratio then ,

(1) The n'th term is given by the formula .

$\sf\longrightarrow\boxed{ a_n=ar^{n-1} }$

(2)Sum of n number of terms ,

$\sf\longrightarrow\boxed{ S_n=\frac{n[r{}^{n}-1]}{r-1}}$

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✺ Therefore :

✏ $\ \ {\dfrac{t_5}{ t_3} =\dfrac{\cancel{a}r^{5-1}}{\cancel{a}r^{3-1}} }$

✏ $\ \ {\dfrac{7}{9}=\dfrac{\cancel{r^4}}{\cancel{r^2}}}$

✏ $\ \ {\dfrac{7}{9}=r^2}$

✏ $\ \ {\boxed{r^2=\dfrac{7}{9}}}$

Hence ,

✏ $\ \ {\dfrac{t_9}{ t_5} =\dfrac{\cancel{a}r^{9-1}}{\cancel{a}r^{5-1}} }$

✏ $\ \ {t_9:t_5=\dfrac{\cancel{r^8}}{\cancel{r^4}}}$

✏ $\ \ {t_9:t_5=r^4}$

✏ $\ \ {t_9:t_5=(r^2)^2}$

✏ $\ \ {t_9:t_5=(\dfrac{7}{9})^2}$

✏ $\ \ {\red{\large{\boxed{t_9:t_5=\dfrac{49}{81}}}}}$

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Answer 2

Answer:-

Let , t(5) = a*r⁴ = 7 -- equation (1)

t(3) = ar² = 9 -- equation (2)

given that,

$\frac{ t_{5} }{ t_{3}} = \frac{7}{9}$

$\frac{a \times {r}^{4} }{a \times {r}^{2} } = \frac{7}{9} \\ \\ {r}^{4} \times {r}^{ - 2} = \frac{7}{9} \\ \\ {r}^{2} = \frac{7}{9}$

Substitute "r²" Value in equation -(1).

ar² = 7

$a \times \frac{7}{9} = 7 \\ \\ a = 7 \times \frac{9}{7} \\ \\ a = 9$

We know that, t(9) = a*r^8=> a*(r²)⁴

and t(5) = a*r⁴ => a*(r²)²

$\frac{t _9 }{t _{5} } = \frac{9 \times (\frac{7}{9}) ^{4} }{9 \times ( { \frac{7}{9} }^{2} )} \\ \\ \frac{ t_{9} }{t _5} = (\frac{7}{9} ) ^{4} - ({ \frac{7}{9} })^{2} \\ \\ \frac{ t_{9} }{ t_{5} } = \frac{ {7}^{2} }{ {9}^{2} } \\ \\ \frac{ t_{9} }{ t_{5} } = \frac{49}{81}$