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(D) Let the $13$ consecutive odd integers be $x, x+2, x+4, \dots, x+24$.The average of the first $7$ integers is given as $37$.The first $7$ integers

Vedclass · Accepted Answer

$43$

Vedclass · Answer

(D) Let the $13$ consecutive odd integers be $x, x+2, x+4, \dots, x+24$.The average of the first $7$ integers is given as $37$.The first $7$ integers are $x, x+2, x+4, x+6, x+8, x+10, x+12$.The average of these $7$ integers is the middle term, which is the $4^{	ext{th}}$ term: $x+6 = 37$.Thus, $x = 31$.The entire series consists of $13$ consecutive odd integers starting from $31$: $31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55$.The average of an arithmetic progression is the middle term. For $13$ terms, the middle term is the $\frac{13+1}{2} = 7^{	ext{th}}$ term.The $7^{	ext{th}}$ term is $x + (7-1) 	imes 2 = 31 + 12 = 43$.Alternatively, the $7^{	ext{th}}$ term is $37 + (7-4) 	imes 2 = 37 + 6 = 43$.

The average of the first $7$ integers in a series of $13$ consecutive odd integers is $37$. What is the average of the entire series?

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