How many consecutive odd integers beginning with 5 will sum to 48… - Vidyadip

Question

How many consecutive odd integers beginning with $5$ will sum to $480?$

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Answer

$5,7,9,11,13, \ldots$
$ S_n=480 $
$ a=5, d=2, S_n=480$
$ S_n=\frac{n}{2}(2 a+(n-1) d)$
$ 480=\frac{n}{2}[2 \times 5+(n-1) 2]$
$ =\frac{n}{2}[10+2 n-2]$
$ 480=\frac{n}{2}[8+2 n]$
$ 8 n+2 n^2=960$
$ 2 n^2+8 n-960=0$
$ \Rightarrow n^2+4 n-480=0$
$ \Rightarrow n^2+24 n-20 n-480=0$
$\Rightarrow n ( n +24)-20( n +24)=0 $
$\Rightarrow( n -20)( n +24)=0 $
$\Rightarrow n =20,-24$
No. of terms cannot be negative.
$\therefore$ No. of consecutive odd integers beginning with $5$ will sum to $480$ is $20.$

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