Question

The ratio of 6th term and 9th term of an
a.P is 7:9.Find the ratio of9th term to 13th term

Answer 1

$\bold{Given:-} \begin{cases} \sf{ \underline{The \: Ratio \: of \: terms \: are:-}} \\ \sf{ {6}^{th} = 7} \\ \sf{ {9}^{th} = 9} \\ \: \: \: \: \: \: \: or \\ \sf{ {6}^{th} \ratio {9}^{th} = 7 \ratio 9} \end{cases}$

${ \underline {\large \star \: {Solution:-}}}$

$\frac{a_6}{a_9} = \frac{7}{9}. \\ \\ \implies \: \frac{a + 5d}{a + 8d} = \frac{7}{9}. \\ \\ \implies \: 9(a + 5d) = 7(a + 8d). \\ \\ \implies \: \: \: \: 9a + 45d = 7a + 56d. \\ \\ \implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 2a = 56d - 45d. \\ \\ \implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: a = \frac{11d}{2} .$

____________________________________

$\implies \: \frac{a_9}{a_{13}} \\ \\ \implies \: \frac{a + 8d}{a + 12d} . \\ \\ \implies \: \frac{ \frac{11d}{2} + 8d }{ \frac{11d}{2} + 12d} . \\ \\ \implies \: \frac{ \frac{11 d+ 16d}{ \cancel2} }{ \frac{11d + 24d}{ \cancel2} }$

$\implies \: \frac{27 \cancel{d}}{35 \cancel{d}} . \\ \\ \implies \: 27\ratio35.$

$\green{ \underline{ \large{ \boxed{Our \: \: Answer}}}} \\ \ \ \\ \green {\implies \: 27 \ratio35.}$

${\underline{ \large{ \star \: {Some \: Information}}}}$

$\purple{\large \boxed{ a_n = a + (n - 1)d}}$