Question

If xth times the xth term of an A.P is equal to n times the yth term , show that (x+y)th term is 0.

please help me ​

Answer 1

${ \bold{ \underline {\underline{To \: prove:}}}}$

Let the first term of AP = a

common difference = d

We have to show that (x+y)th term is zero or a + (x+y-1)d = 0

mth term = a + (x-1)d

nth term = a + (y-1) d

${ \bold{ \underline {\underline{Given \: that:}}}}$

m{a +(x-1)d} = n{a + (y -1)d}

⇒ ax + x²d -xd = ay + y²d - yd

⇒ ax - ay + x²d - y²d -xd + yd = 0

⇒ a(x-y) + (x²-y²)d - (x-y)d = 0

⇒ a(x-y) + {(x-y)(x+y)}d -(x-y)d = 0

⇒ a(x-y) + {(x-y)(x+y) - (x-y)} d = 0

⇒ a(x-y) + (x-y)(x+y-1) d = 0

⇒ (x-y){a + (x+y-1)d} = 0

⇒ a + (x+y -1)d = 0/(x-y)

⇒ a + (x+y -1)d = 0

${ \bold{ \pink{ \underline {\underline{Hope \: it \: helps \: you}}}}}$