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tatiyna
3 years ago
8

2. Determine the sum of the first 400 ODD numbers.​

Mathematics
1 answer:
il63 [147K]3 years ago
3 0

Odd numbers take the form 2n-1, where n\ge1 is an integer. When n=400, the last odd number would be 799. So we're adding

S=1+3+5+\cdots+795+797+799

By reversing the order of terms, we have

S=799+797+795+\cdots+5+3+1

and we can pair up terms in both sums at the same position to write

2S=(1+799)+(3+797)+(5+795)+\cdots(795+5)+(797+3)+(799+1)

so that we are basically adding 400 copies of 800, and from there we can find the value of the sum right away:

2S=400\cdot800\implies S=160,000

###

We could also make use of the formulas,

\displaystyle\sum_{i=1}^n1=n

\displaystyle\sum_{i=1}^ni=\dfrac{n(n+1)}2

We have

S=\displaystyle\sum_{i=1}^{400}(2i-1)=2\sum_{i=1}^{400}i-\sum_{i=1}^{400}1=400(400+1)-400=400^2=160,000

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