Question

Ifp th ,q th, r th term of an A.P. is x, y and z respectively. Show that x (q-r) + y
(r-p) + z(p-q) = 0.

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Answer 1

Answer:

Hey there !!

→ Given:-

Pth term = x.

Qth term = y.

And, Rth term = z.

→ To prove :-

=> x( q - r ) + y( r - p ) + z( p - q ) = 0.

→ Solution:-

Let a be the first term and D be the common difference of the given AP. Then,

T\tiny pp = a + ( p - 1 )d.

T\tiny qq = a + ( q - 1 )d.

And,

T\tiny rr = a + ( r - 1 )d.

▶ Now,

=> a + ( p - 1 )d = x..........(1).

=> a + ( q - 1 )d = y..........(2).

=> a + ( r - 1 )d = z...........(3).

▶ On multiplying equation (1) by ( q - r ), (2) by ( r - p ) and (3) by ( p - q ), and adding, we get

=> x( q - r ) + y( r - p ) + z( p - q ) = x•{( q - r ) + ( r - p ) + ( p - q )} + d•{( p - 1 ) ( q - r ) + ( q - 1 ) ( r - p ) + ( r - 1 ) ( p - q )}

=> x( q - r ) + y( r - p ) + z( p - q ) = ( x × 0 ) + ( d × 0 ).

⇒ x( q - r ) + y( r - p ) + z( p - q ) = 0.

✔✔ Hence, it is proved ✅✅.

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