Math, asked by kritibudhrain27, 3 months ago

if a,b,c be respectively the Xth, Yth and Zth terms of an AP prove that a(y-z) + b(z-x) + c( x-y)=0​

Answers

Answered by snehitha2
7

Step-by-step explanation :

Given,

a,b,c be respectively the xth, yth and zth terms of an AP

To prove,

a(y - z) + b(z - x) + c(x - y) = 0​

Solution,

nth term of an AP is given by,

tₙ = t + (n - 1)d

where

t is the first term

d is the common difference

  • xth term of AP = a

  a = t + (x - 1)d --[1]

  • yth term of AP = b

 b = t + (y - 1)d --[2]

  • zth term of AP = c

c = t + (z - 1)d --[3]

equation [1] - equation [2]

a - b = t + (x - 1)d - [t + (y - 1)d]

a - b = t + xd - d - [t + yd - d]

a - b = t + xd - d - t - yd + d

a - b = xd - yd

equation [2] - equation [3]

b - c = t + (y - 1)d - [t + (z - 1)d]

b - c = t + yd - d - [t + zd - d]

b - c = t + yd - d - t - zd + d

b - c = yd - zd

equation [3] - equation [1]

c - a = t + (z - 1)d - [t + (x - 1)d]

c - a = t + zd - d - [t + zx - d]

c - a = t + zd - d - t - zx + d

c - a = zd - xd

we have to prove a(y - z) + b(z - x) + c(x - y) = 0​

LHS = a(y - z) + b(z - x) + c(x - y)

       = ay - az + bz - bx + cx - cy

       = ay - cy - az + bz - bx + cx

       = y(a - c) + z(b - a) + x(c - b)

       = -y(c - a) - z (a - b) - x (b - c)

       = -y (zd - xd) - z(xd - yd) - x(yd - zd)

       = -yzd + xyd - zxd + yzd - xyd + zxd

       = -yzd + yzd + xyd - xyd + zxd - zxd

       = 0

       = RHS

∴ LHS = RHS

a(y - z) + b(z - x) + c(x - y) = 0​

Hence proved!

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