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Find the sum:$1 + 3 + 5 + 7 + …….. + 199$
Given:
Given sequence is $1 + 3 + 5 + 7 + …….. + 199$.
To do:
We have to find the sum of $1 + 3 + 5 + 7 + …….. + 199$.
Solution:
Here,
\( 1+3+5+7+\ldots+199 \) is in A.P.
\( a=1, d=3-1=2 \) and \( l=199 \)
We know that,
\( a_{n}=a+(n-1) d \)
\( \Rightarrow 199=1+(n-1) \times 2 \)
\( \Rightarrow 199=1+2 n-2 \)
\( \Rightarrow 2 n=199-1+2=200 \)
\( n=\frac{200}{2}=100 \)
\( S_{n}=\frac{n}{2}[a+l]=\frac{100}{2}(1+199) \)
\( =\frac{100}{2} \times 200=10000 \)
Therefore, the sum of the given sequence is $10000$. 
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