if the numbers a,b,c,d and e are in an A.P. then prove that a-4b+6c-4d+c=0
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a, b , c , d and e are in AP
let a = x
b = x + d
c = x + 2d
d = x +3d
e = x +4d
put this in LHS
LHS = a -4b +6c -4d +e
= x -4(x +d ) +6(x +2d ) -4(x +3d) +x +4d
={ x -4x +6x -4x +x} +{ -4d +12d -12d +4d}
= 0 + 0 = 0 = RHS
let a = x
b = x + d
c = x + 2d
d = x +3d
e = x +4d
put this in LHS
LHS = a -4b +6c -4d +e
= x -4(x +d ) +6(x +2d ) -4(x +3d) +x +4d
={ x -4x +6x -4x +x} +{ -4d +12d -12d +4d}
= 0 + 0 = 0 = RHS
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