Let T r be the rᵗʰ term of an AP, where the first term is a and common…
2019
Let T r be the rᵗʰ term of an AP, where the first term is a and common difference is d. If for some positive integers m ≠ n, T m = 1/n and T n = 1/m, then a − d equals:
- A.
0
- B.
1/mn
- C.
1/m + 1/n
- D.
1
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Correct answer: A
Given T_r = a + (r - 1)d. We are told that T_m = 1/n and T_n = 1/m for positive integers m ≠ n.
Write the two equations:
a + (m - 1)d = 1/n
a + (n - 1)d = 1/m
Subtract the second equation from the first to eliminate a:
(m - n)d = 1/n - 1/m = (m - n)/(mn)
Since m ≠ n, divide both sides by (m - n) to get d = 1/(mn).
Substitute d = 1/(mn) into a + (m - 1)d = 1/n to find a:
a = 1/n - (m - 1)/(mn) = (m/(mn)) - ((m - 1)/(mn)) = 1/(mn)
Therefore a = 1/(mn) and d = 1/(mn), so a − d = 0.
Answer: a − d = 0
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