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The $20^{t h}$ term of an arithmetic sequence is $\log_{10} 20 and the 32^{nd}$ term is $\log_{10} 32$ . If exactly one term of the arithmetic sequence is a rational number which in lowest form is $\frac{p}{q} (p, q \in N) then (p + q) = . . . . . .$

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answer is 9.

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Detailed Solution

Given

$a + 19 d = l o g_{10} 20 . . . . . (1)$

$a + 31 d = l o g_{10} 32 . . . . (2)$

$(2) - (1)$

$12 d = \log_{10} \frac{32}{20} = \log_{10} 16 - 1$

$\log_{10} 2 = \frac{12 d + 1}{4} \dots (A)$

$\log_{10} 2 = \frac{2 a + 50 d - 1}{6} \dots (B)$

$\frac{a + 64 d}{17^{th} term} = \frac{5}{4}$

Hence, $17^{t h}$ term is rational and its value is $\frac{5}{4} = \frac{p}{q} \Rightarrow (p + q) = 9$

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