The sum of first eight terms of an arithmetic progression is 100. The ratio of its $9^{t h}$ term to its $19^{t h}$ term is 13 : 28. The thirteenth term of the AP is

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Solution

The correct option is B 38
Given: $S_{8} = 100$ and $\frac{a_{9}}{a_{19}} = \frac{13}{28}$
Let the first term be a and the common difference be d.
Now, $S_{8} = 100$
$\Rightarrow \frac{8}{2} [2 a + (8 - 1) d] = 100$
$\Rightarrow 4 [2 a + 7 d] = 100$
$\Rightarrow 2 a + 7 d = 25$ ............(i)
Also, $\frac{a_{9}}{a_{19}} = \frac{13}{28}$
$\Rightarrow \frac{a + (9 - 1) d}{a + (19 - 1) d} = \frac{13}{28}$
$\Rightarrow \frac{a + 8 d}{a + 18 d} = \frac{13}{28}$
$\Rightarrow 28 a + 224 d = 13 a + 234 d$
$\Rightarrow 28 a - 13 a = 234 d - 224 d$
$\Rightarrow 15 a = 10 d$
$\Rightarrow a = \frac{2}{3} d$ ..............(ii)
Substituting the value of a form (ii) and (i), we get
$2 \times \frac{2 d}{3} + 7 d = 25$
$\Rightarrow \frac{4 d + 21 d}{3} = 25$
$\Rightarrow 25 d = 75$
$\Rightarrow d = 3$
From (ii), we get
$a = \frac{2}{3} \times 3$
$\Rightarrow a = 2$
$∴ 13^{t h}$ term $a_{13} = a + (13 - 1) d$
$= a + 12 d$
$= 2 + 12 \times 3$
$= 2 + 36$
$= 38$
Hence the correct answer is option (2)

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