The ratio of the 7^{th} to the 3^{rd} term of | vedclass

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(D) Let $a$ be the first term and $d$ be the common difference of the $A.P.$The $n^{th}$ term of an $A.P.$ is given by $a_n = a + (n-1)d$.Given that t

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$10:3$

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(D) Let $a$ be the first term and $d$ be the common difference of the $A.P.$The $n^{th}$ term of an $A.P.$ is given by $a_n = a + (n-1)d$.Given that the ratio of the $7^{th}$ term to the $3^{rd}$ term is $12:5$:$\frac{a + 6d}{a + 2d} = \frac{12}{5}$Cross-multiplying gives:$5(a + 6d) = 12(a + 2d)$$5a + 30d = 12a + 24d$$30d - 24d = 12a - 5a$$6d = 7a$$a = \frac{6}{7}d$Now,we need to find the ratio of the $13^{th}$ term to the $4^{th}$ term:$\frac{a_{13}}{a_4} = \frac{a + 12d}{a + 3d}$Substituting $a = \frac{6}{7}d$ into the expression:$\frac{\frac{6}{7}d + 12d}{\frac{6}{7}d + 3d} = \frac{\frac{6d + 84d}{7}}{\frac{6d + 21d}{7}} = \frac{90d}{27d} = \frac{90}{27}$Simplifying the fraction by dividing by $9$:$\frac{90}{27} = \frac{10}{3}$Thus,the ratio is $10:3$.

The ratio of the $7^{th}$ to the $3^{rd}$ term of an $A.P.$ is $12:5$. Find the ratio of the $13^{th}$ to the $4^{th}$ term.

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