8. Sequences and Series
hard

જો એક સમાંતર શ્રેણી $a_{1} a_{2}, a_{3}, \ldots$ ના પ્રથમ $11$ પદોનો સરવાળો $0\left(\mathrm{a}_{1} \neq 0\right)$ થાય અને સમાંતર શ્રેણી $a_{1}, a_{3}, a_{5}, \ldots, a_{23}$ પદોનો સરવાળો $k a_{1}$ થાય તો $k$ ની કિમત મેળવો 

A

$\frac{121}{10}$

B

$-\frac{72}{5}$

C

$\frac{72}{5}$

D

$-\frac{121}{10}$

(JEE MAIN-2020)

Solution

$a_{1}+a_{2}+a_{3}+\ldots \ldots+a_{11}=0$

$\Rightarrow\left(a_{1}+a_{11}\right) \times \frac{11}{2}=0$

$\Rightarrow \mathrm{a}_{1}+\mathrm{a}_{11}=0$

$\Rightarrow \mathrm{a}_{1}+\mathrm{a}_{1}+10 \mathrm{d}=0$

where d is common difference

$\Rightarrow \quad \mathrm{a}_{1}=-5 \mathrm{d}$

$a_{1}+a_{3}+a_{5}+\ldots \ldots+a_{23}$

$=\left(a_{1}+a_{23}\right) \times \frac{12}{2}=\left(a_{1}+a_{1}+22 d\right) \times 6$

$=\left(2 a_{1}+22\left(\frac{-a_{1}}{5}\right)\right) \times 6$

$=-\frac{72}{5} a_{1} \Rightarrow K=\frac{-72}{5}$

Standard 11
Mathematics

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