Math, asked by ipsitamoody1998, 2 months ago

VAHTH
leading Institute for Banking, Railway & SSC
1. The ratio of the 7th to the 3rd term of an AP is 12:5. Find the ratio of 13th to the 4th term.
(a) 8:5 (b) 9:4 (c) 7:3
(d) 10:3
th​

Answers

Answered by MrImpeccable
7

ANSWER:

Given:

  • Ratio of 7th term to 3rd term in an AP = 12:5

To Find:

  • Ratio of 13th term to the 4th term.

Solution:

\text{We are given that,}\\\\:\longrightarrow a_7:a_3=12:5\\\\:\implies\dfrac{a_7}{a_3}=\dfrac{12}{5}\\\\\text{We know that,}\\\\:\hookrightarrow a_n=a+(n-1)d\\\\\text{where, a is first term, n is no. of term and d is common difference.}\\\\\text{So,}\\\\:\implies\dfrac{a_7}{a_3}=\dfrac{12}{5}\\\\:\implies\dfrac{a+(7-1)d}{a+(3-1)d}=\dfrac{12}{5}\\\\:\implies\dfrac{a+6d}{a+2d}=\dfrac{12}{5}\\\\\text{On cross-multiplying,}\\\\:\implies5(a+6d)=12(a+2d)\\\\:\implies5a+30d=12a+24d\\\\\text{Transposing 5a to RHS and 24d to LHS,}\\\\:\implies30d-24d=12a-5a\\\\:\implies6d=7a\\\\\text{Transposing 6 to RHS,}\\\\:\implies d=\dfrac{7a}{6}- - - -(1)

\text{We need to find,}\\\\:\longrightarrow a_{13}:a_4\\\\:\implies\dfrac{a_{13}}{a_4}\\\\:\implies\dfrac{a+(13-1)d}{a+(4-1)d}\\\\:\implies\dfrac{a+12d}{a+3d}\\\\\text{Using (1),}\\\\:\implies\dfrac{a+12\!\!\!\!/^{\:2}\left(\frac{7a}{6\!\!\!/}\right)}{a+3\!\!\!/\left(\frac{7a}{6\!\!\!/_{\:2}}\right)}\\\\:\implies\dfrac{a+2(7a)}{a+1\left(\frac{7a}{2}\right)}\\\\:\implies\dfrac{a+14a}{\frac{2a+7a}{2}}\\\\:\implies\dfrac{2(15a)}{9a}\\\\:\implies\dfrac{30a\!\!\!/}{9a\!\!\!/}\\\\:\implies\dfrac{30\!\!\!\!/^{\:10}}{9\!\!\!/_{\:3}}\\\\:\implies\dfrac{10}{3}\\\\:\implies10:3\\\\\bf{:\implies a_{13}:a_4=10:3}

Formula Used:

  • General term = a_n=a+(n-1)d
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