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Prove that 1+2+3+.....n=(n(n+1))/(2)...

Prove that `1+2+3+.....n=(n(n+1))/(2)`

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Let `P (n): 1+2+3+.....n=((n+1))/(2)` be the given statement
Step 1 : Putt n=1
Then , LHS=1 and RSH `=(1(1+1))/(2)=1`
`:. LHS=RHS rArrP(n)` is true for n=1
Step 2: Assume that P(n) is true for n=x
`:. 1+2+3+.....k=(k(k+1))/(2)`
Adding (k+1) on both sides, we get
`1+2+3+.....+k+(k+1)=((k+1))/(2)+(k+1)`
`=(k+1)((k)/(2)+1)=((k+1)(k+2))/(2)=((k+1)(bar(k+1)+1))/(2)`
`rArr P(n)` is true for `n=k+1`
`:.` By the principle of mathcmatical induction P(n) is true fo all natural numbers n.
Hence ,`+2+3+.....+n(n(n+1))/(2)"for all " n in N`
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