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

Let's break this problem down to understand how consecutive numbers work and how to connect the two sequences.

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When we have 5 consecutive odd numbers, they follow a pattern where each number is 2 more than the previous one.

If we call the first odd number $a_1$, then:

$a_1 = a_1$ (first number)

$a_2 = a_1 + 2$ (second number)

$a_3 = a_1 + 4$ (third number)

$a_4 = a_1 + 6$ (fourth number)

$a_5 = a_1 + 8$ (fifth number)

This means $a_3$ is the middle number of our odd sequence.

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We're told that the new sequence consists of 5 consecutive even numbers ending with $2a_3$.

Since consecutive even numbers also differ by 2, and the sequence ends with $2a_3$, the five numbers are:

$2a_3 - 8$ (first even number)

$2a_3 - 6$ (second even number)

$2a_3 - 4$ (third even number)

$2a_3 - 2$ (fourth even number)

$2a_3$ (fifth even number)

If the last number is $2a_3$, then working backwards by subtracting 2 each time gives us the previous even numbers.

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The sum of these 5 consecutive even numbers equals 450:

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

Collecting like terms:

$2a_3 + 2a_3 + 2a_3 + 2a_3 + 2a_3 - 8 - 6 - 4 - 2 = 450$

$10a_3 - 20 = 450$

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$10a_3 - 20 = 450$

$10a_3 = 470$

$a_3 = 47$

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Now that we know $a_3 = 47$, we can find $a_5$.

Since $a_3$ is the middle number of our consecutive odd sequence, and consecutive odd numbers differ by 2:

$a_4 = a_3 + 2 = 47 + 2 = 49$

$a_5 = a_4 + 2 = 49 + 2 = 51$

Our complete odd sequence is: $43, 45, 47, 49, 51$

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Therefore, $a_5 = 51$