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 :

Question:

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 :

  1. $\frac{1}{4}$

  2. $\frac{1}{5}$

  3. $\frac{1}{7}$

  4. $\frac{1}{6}$


Correct Option: , 4

Solution:

Sum of 1 st 25 terms $=$ sum of its next 15 terms

$\Rightarrow\left(\mathrm{T}_{1}+\ldots . .+\mathrm{T}_{25}\right)=\left(\mathrm{T}_{26}+\ldots . .+\mathrm{T}_{40}\right)$

$\Rightarrow\left(\mathrm{T}_{1}+\ldots . .+\mathrm{T}_{40}\right)=2\left(\mathrm{~T}_{1}+\ldots . .+\mathrm{T}_{25}\right)$

$\Rightarrow \frac{40}{2}[2 \times 3+(39 \mathrm{~d})]=2 \times \frac{25}{2}[2 \times 2+24 \mathrm{~d}]$

$\Rightarrow \mathrm{d}=\frac{1}{6}$

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