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8. Sequences and Series
hard
If the first term of an $A.P.$ is $3$ and the sum of its first $25$ terms is equal to the sum of its next $15$ terms, then the common difference of this $A.P.$ is :
A
$\frac{1}{4}$
B
$\frac{1}{5}$
C
$\frac{1}{7}$
D
$\frac{1}{6}$
(JEE MAIN-2020)
Solution
Sum of 1 st 25 terms $=$ sum of its next 15 termss
$\Rightarrow\left( T _{1}+\ldots \ldots+ T _{25}\right)=\left( T _{26}+\ldots . .+ T _{40}\right)$
$\Rightarrow\left( T _{1}+\ldots . .+ T _{40}\right)=2\left( T _{1}+\ldots \ldots+ T _{25}\right)$
$\Rightarrow \frac{40}{2}[2 \times 3+(39 d )]=2 \times \frac{25}{2}[2 \times 2+24 d ]$
$\Rightarrow d=\frac{1}{6}$
Standard 11
Mathematics
