Maths / Arithmetic Progression / General Term Of an AP

QUESTION
 

If pth ,qth and rth terms of an AP are a, b and c respectively, show that a(q-r)+b(r-p)+c(p-q)=0.

EXPLANATION
Explain TypeExplanation Content
Text

large a_p=a

large x-(p-1)d=a

Multiplying both sides with (q-r)

large x(q-r)-(p-1)(q-r)d=a(q-r)

large xq-xr-(pq-pr-q+r)d=a(q-r)                    (1)

large a_q=b

large x-(q-1)d=b

Multiplying both sides with (r-p)

large x(r-p)-(q-1)(r-p)d=b(r-p)

large xr-xp-(qr-qp-r+p)d=b(r-p)                    (2)

large a_r=c

large x-(r-1)d=c

Multiplying both sides with (p-q)

large x(p-q)-(r-1)(p-q)d=c(p-q)

large xp-xq-(rp-rq-p+q)d=c(p-q)                    (3)

adding 1, 2 and 3 we get

large xq-xr-(pq-pr-q+r)d+xr-xp-(qr-qp-r+p)d+xp-xq-(rp-rq-p+q)d=a(q-r)+b(r-p)+c(p-q)

large (xq-xr+xr-xp+xp-xq)-(pq-pr-q+r+qr-qp-r+p+rp-rq-p+q)d=a(q-r)+b(r-p)+c(p-q)

large (0)-(0)d=a(q-r)+b(r-p)+c(p-q)

large a(q-r)+b(r-p)+c(p-q) =0

 

 

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