Question

plz solve this as soon as possible​

Answer 1

Step-by-step explanation:

Given :-

In an A.P. , Tp = q and Tq = p

To find:-

Find the rth term of the A.P. ?

Solution :-

We know that

nth term of the AP is Tn = a+(n-1)d

Where, a = First term

d = Common difference

• n = number of terms

Given that

pth term of the A.P. = Tp = q

=> a + (p-1)d = q

=> (p-1)d = q - a

=> d = (q-a)/(p-1) -----------(1)

and

qth term of the A.P. = Tq = p

=> a + (q-1) d = p

=> (q-1)d = p-a

=> d = (p-a)/(q-1) ----------(2)

From (1) &(2)

=> (q-a)/(p-1) = (p-a)/(q-1)

On applying cross multiplication then

=> (q-a)×(q-1) = (p-a)×(p-1)

=> q²-q-aq+a = p²-p-ap+a

=> q²-q-aq = p²-p-ap

=> ap-aq = p²-p+q-q²

=> a(p-q) = (p²-q²)-(p-q)

=> a(p-q) = (p+q)(p-q)-(p-q)

=> a(p-q) = (p-q)[(p+q)-1]

On cancelling (p-q) both sides then

=> a = (p+q-1) -----------(3)

On Substituting the value of a in (2) then

=> d = [p-(p+q-1)]/(q-1)

=> d = (p-p-q+1)/(q-1)

=> d = (-q+1)/(q-1)

=> d = -(q-1)/(q-1)

=> d = -1

We have,

a = p+q-1 and d = -1

Now, rth term of the AP

=> Tr = a + (r-1)d

=> Tr = (p+q-1)+(r-1)(-1)

=> Tr = p+q-1+(-r+1)

=> Tr = p+q-1-r+1

=> Tr = p+q-r

Therefore, Tr = p+q-r

Answer:-

rth term of the given A.P. is p+q-r

Used formulae:-

• nth term of the AP is Tn = a+(n-1)d
• Where, a = First term
• d = Common difference
• n = number of terms