The sum of first $7$ terms is $63$ and sum of next of seven terms is $161$ find $25^{t h}$ terms

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Solution

$S_{7} = 63$
$\frac{7}{2} [2 a + (7 - 1) d] = 63$
$14 a + 42 d = 126$
$a + 3 d = 9$ --- (i)
Sum of next seven term is $161$
$S_{14} - S_{7} = 161$
$\frac{14}{2} [2 a + 13 d] - 63 = 161$
$7 (2 a + 13 d) = 224$
$a + \frac{13}{2} d = 16$ ---- (ii)
on subtracting (i) from (ii)
we get,
$d = 2$
from (i)
$a = 3$
$a_{25} = a + 24 d$
$= 3 + 24 \times 2 = 51$

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The sum of the first $7$ terms of an AP is $63$ and the sum of it's next $7$ terms is $161$ . Find the $28^{t h}$ term of this AP.

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