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Let $a_{1},a_{2},a_{3},a_{4},a_{5}$ be a sequence of five consecutive odd numbers. Consider a new sequence of five consecutive even numbers ending with $2a_{3}$. If the sum of the numbers in the new sequence is $450$, then $a_{5}$ is 

  1. $50$
  2. $51$
  3. $52$
  4. $49$
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Given that,   $a_1,a_2,a_3,a_4,a_5$ are five consecutive odd numbers.

  •  $a_3-4,a_3-2,{\color{Red} {a_3}},a_3+2,a_3+4$

A new sequence of five consecutive even numbers ending with $2a_3$ will be :

  • $2a_3-8, 2a_3-6, 2a_3-4,2a_3-2,2a_3$

The sum of the numbers in the new sequence $=450$

$\Rightarrow (2a_3-8)+(2a_3-6)+(2a_3-4)+(2a_3-2)+(2a_3)=450$

$\Rightarrow 10a_3-20 = 450$

$\Rightarrow 10a_3 = 470$

$\Rightarrow \boxed{a_3 = 47}$

$\Rightarrow a_5 = a_3+4=47+4=51.$

Correct Answer $:\text{B}$

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