if pth term of a AP is q and qth term is p, prove that the rth term is p+q-r which term of the series is zero
Answers
Answer :
(p+q)ᵗʰ term is 0
Step-by-step explanation :
Given,
- pth term of AP = q
- qth term of AP = p
To find,
- the rth term of AP
- which term is zero
Solution,
nth term of AP is given by,
aₙ = a + (n - 1)d
where
a is the first term
d is the common difference
pᵗʰ term :
aₚ = a + (p - 1)d
q = a + pd - d ---[1]
qᵗʰ term :
a(q) = a + (q - 1)d
p = a + qd - d ---[2]
Subtract equation [2] from equation [1],
q - p = a + pd - d - (a + qd - d)
q - p = a + pd - d - a - qd + d
q - p = pq - qd
q - p = (p - q)d
-(p - q) = (p - q)d
d = -(p - q)/(p - q)
d = -1
⤳ Common difference = -1
Substitute d = -1 in equation [1]
q = a + pd - d
q = a + p(-1) - (-1)
q = a - p + 1
a = p + q - 1
⤳ first term = p + q - 1
we have to find the rᵗʰ term
put n = r
aᵣ = a + (r - 1)d
aᵣ = p + q - 1 + (r - 1)(-1)
aᵣ = p + q - 1 - r + 1
aᵣ = p + q - r
⤳ rth term = p + q - r
Hence proved!
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Let nth term be zero,
put aₙ = 0
0 = a + (n - 1)d
0 = p + q - 1 + (n - 1)(-1)
0 = p + q - 1 - n + 1
0 = p + q - n
n = p + q
Therefore, (p+q)ᵗʰ term is 0