Which of the following are harmonic progressions-a 1- 1-3- 1-5- 1-7- -b 1- 1-3- 1-6- 1-12- - -c 1- 2-3- 1-2- 2-5- - -d 1-2- 1-5- 1-10- 1-17- - -

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Standard X

Mathematics

Arithmetic Progression

Which of the ...

Question

Which of the following are harmonic progressions.

(a)

(b)

(c)

(d)

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Solution

(a) The given sequence is

Taking reciprocal of every term, we get 1, 3, 5, 7 …

These terms form an A.P. with and .

Thus, is in H.P.

(b) The given sequence is

Taking reciprocal of every term, we get 1, 3, 6, 12 ….

These terms do not form an A.P. [3 − 1 = 2, 6 − 3 = 3]

Thus, is not in H.P.

(c) The given sequence is

Taking reciprocal of every term, we get .

These terms form an A.P with and .

Thus, is in H.P.

(d) The given sequence is

Taking reciprocal of every term, we get 2, 5, 10, 17 …..

These terms do not form an A.P. [5 − 2 = 3, 10 − 5 = 5].

Thus, is not in H.P.

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(ii)

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(iii)

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(iv)

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Which of the following are harmonic progressions. (a) (b) (c) (d)

Which of the following are harmonic progressions.

(a)

(b)

(c)

(d)