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in Arithmetic Progression by (70.9k points)
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The first , second and the last terms of an A.P. are `a ,b , c` respectively. Prove that the sum is `((a+c)(b+c-2a))/(2(b-a))` .

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`1^(st)` term=a
`2^(nd)`term=b
`3^(rd)`term=c
Common diff,d=(b-a)
c=a+(n-1)(b-a)
c-a=(n-1)(b-a)
`n-1=(c-a)/(b-a)`
`n=(c-a)/(b-a)+1`
`n=(c-a+b-a)/(b-a)=(b+c-2a)/(b-a)`
`S=n/2(a+l)`
`S=(b+c-2a)/(2(b-a))*(a+c)`
`S=((a+c)(b+c-2a))/(2(b-a))`.

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