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# In an A.P., the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20 th term is minus 112.

3.  In an A.P., the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is –112.

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First term =a=2

Let the series be $2,2+d,2+2d,2+3d,.......................$

Sum of first five terms $=10+10d$

Sum of next five terms $=10+35d$

Given : The sum of the first five terms is one-fourth of  the next five terms.

$10+10d=\frac{1}{4}(10+35d)$

$\Rightarrow \, \, 40+40d=10+35d$

$\Rightarrow \, \, 40-10=35d-40d$

$\Rightarrow \, \, 30=-5d$

$\Rightarrow \, \, d=-6$

To prove  : $a_2_0=-112$

L.H.S : $a_2_0=a+(20-1)d=2+(19)(-6)=2-114=-112=R.H.S$

Hence, 20th term is –112.

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