Question

the sum of first 20 terms of an AP . is one third of the sum of next 20 terms. if first term is 1 find the sum of first 30 terms of this AP

Answer 1

sum of first 30 terms is 900

Answer 2

Let the first term be 1 and common difference be d.

★According to the question,

⠀⠀⠀

$:\implies\sf S_{20} = \frac{1}{3} \bigg( S_{40} -S_{20} \bigg)$⠀⠀

$:\implies\sf S_{20} = \frac{1}{3} \times \bigg(S_{20} \: \bigg)$

$:\implies\sf \frac{20}{2} \bigg(2 + 19d \bigg) = \frac{1}{3} \bigg( \frac{40}{2} (2 + 39d)\bigg)$

$:\implies\sf {\cancel {10}} (2+19d)= \frac{{\cancel {10}}}{3}(2 + 59d)$

$:\implies\sf 6+59d=2+59d$

$:\implies\sf 6-2+ {\cancel{59d}} - {\cancel{59d}} =0$

$:\implies\sf 4=2d$

$:\implies\sf d= {\cancel \dfrac{4}{2}}$

$:\implies{\boxed{\frak{\purple{d=2}}}}\;\bigstar$

Now,

$:\implies\sf S_30= \dfrac {30}{2} (2+29×2)$

$:\implies\sf S_30= 15×60$

$:\implies{\boxed{\frak{\pink{900}}}}\;\bigstar$

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