8. Sequences and Series
hard

माना $\mathrm{x}_1, \mathrm{x}_2 \ldots, \mathrm{x}_{100}$ एक समांतर श्रेणी में हैं, जिनका माध्य 200 है तथा $x_1=2$ है। यदि $y_i=i\left(x_i-i\right), 1 \leq i \leq 100$ हैं, तो $\mathrm{y}_1, \mathrm{y}_2, \ldots \ldots, \mathrm{y}_{100}$ का माध्य है

A

$10101.50$

B

$10051.50$

C

$10049.50$

D

$10100$

(JEE MAIN-2023)

Solution

$\text { Mean }=200$

$\Rightarrow \frac{\frac{100}{2}(2 \times 2+99 d)}{100}=200$

$\Rightarrow 4+99 d =400$

$\Rightarrow d=4$

$y_i=i(x i-i)$

$=i(2+(i-1) 4-i)=3 i^2-2 i$

$\text { Mean }=\frac{\sum y_i}{100}$

$=\frac{1}{100} \sum \limits_{i=1}^{100} 3 i^2-2 i$

$=\frac{1}{100}\left\{\frac{3 \times 100 \times 101 \times 201}{6}-\frac{2 \times 100 \times 101}{2}\right\}$

$=101\left\{\frac{201}{2}-1\right\}=101 \times 99.5$

$=10049 \cdot 50$

Standard 11
Mathematics

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