Question

If pth,qth,rth them of an A.P
respectivly are a,b,c Then find the value of a(q-r)+b(r-p)+c(p-q)​

Answer 1

Answer:

☆c.(p-q)d+a.(q-r)d+b.(r-p)d=0

Step-by-Step explanation:

Let,

the first term=A

Common Difference=d

pth term=a

qth term=b

rth term=c

a=A+(p-1)d ----------- i

b=A+(q-1)d ----------- i

c=A+(r-1)d -----------iii

Subtracting equation ii from i , iii from ii ,i from iii

a-b=(p-q)d ----------- iv

b-c=(q-r)d ------------ v

c-a=(r-p)d ------------ vi

Multiplying equation iv by c, v by a and vi by b

(a-b).c=c.(p-q)d ----------- vii

(b-c).a=a.(q-r)d ------------ viii

(c-a).b=b.(r-p)d ------------ix

Adding equation vii,viii and ix

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

ac-bc+ab-ac+bc-ab=c.(p-q)d+a.(q-r)d+b.(r-p)d

ac-ac+bc-bc+ab-ab=c.(p-q)d+a.(q-r)d+b.(r-p)d

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

☆c.(p-q)d+a.(q-r)d+b.(r-p)d=0

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Answer 2

Answer:

☆c.(p-q)d+a.(q-r)d+b.(r-p)d=0

Step-by- explanation:

Let,

the first term=A

Common Difference=d

pth term=a

qth term=b

rth term=c

a=A+(p-1)d ----------- i

b=A+(q-1)d ----------- ii

c=A+(r-1)d -----------iii

Subtracting equation ii from i , iii from ii ,i from iii

a-b=(p-q)d ----------- iv

b-c=(q-r)d ------------ v

c-a=(r-p)d ------------ vi

Multiplying equation iv by c, v by a and vi by b

(a-b).c=c.(p-q)d ----------- vii

(b-c).a=a.(q-r)d ------------ viii

(c-a).b=b.(r-p)d ------------ix

Adding equation vii,viii and ix

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

ac-bc+ab-ac+bc-ab=c.(p-q)d+a.(q-r)d+b.(r-p)d

ac-ac+bc-bc+ab-ab=c.(p-q)d+a.(q-r)d+b.(r-p)d

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

☆c.(p-q)d+a.(q-r)d+b.(r-p)d=0

Hope it helps you.

Please mark it as brainlist.