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The ratio of the $10^{th}$ term to its $30^{th}$ term of an A.P. is $1: 3$ and the sum of its first six terms is $42$. Find the first term and the common difference of A.P.
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Let $a$ be the first term and $d$ be the common difference.
$\frac{a + 9d}{a + 29d} = \frac{1}{3}$
$\Rightarrow a=d \dots (i)$
$\frac{6}{2}(2a + 5d) = 42$
$\Rightarrow 2a + 5d = 14 \dots (ii)$
Solving (i) and (ii)
$a=2$ and $d=2$
$\frac{a + 9d}{a + 29d} = \frac{1}{3}$
$\Rightarrow a=d \dots (i)$
$\frac{6}{2}(2a + 5d) = 42$
$\Rightarrow 2a + 5d = 14 \dots (ii)$
Solving (i) and (ii)
$a=2$ and $d=2$