If three real numbers a, b, c are in harmonic progression, then which of the following is true ?

Question

If three real numbers a, b, c are in harmonic progression, then which of the following is true ? (A) (1/a) , b , (1/c) are in AP (B) (1/bc), (1/ca), (1/ab) are in HP (C) ab, bc, ca are in HP (D) (1/b), (1/c), (1/a) are in HP

Tardigrade Admin · Accepted Answer

Since, a, b, c are in HP. ∴ (1/a), (1/b), (1/c) are in AP . ⇒ (abc/a), (abc/b), (abc/c) are in AP. ⇒ bc, ca, ab are in AP. ⇒ (1/bc) , (1/ca), (1/ab) are in HP.

Q. If three real numbers $a, b, c$ are in harmonic progression, then which of the following is true ?

$\frac{1}{a}, b, \frac{1}{c}$ are in $A P$

$\frac{1}{b c}, \frac{1}{c a}, \frac{1}{ab}$ are in $H P$

$ab, b c, c a$ are in $H P$

$\frac{1}{b}, \frac{1}{c}, \frac{1}{a}$ are in $H P$

Since, $a, b, c$ are in $H P$ .

$∴ \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in $A P$ .

$\Rightarrow \frac{ab c}{a}, \frac{ab c}{b}, \frac{ab c}{c}$ are in $A P$ .

$\Rightarrow b c, c a, ab$ are in $A P$ .

$\Rightarrow \frac{1}{b c}, \frac{1}{c a}, \frac{1}{ab}$ are in $H P$ .

Q. If three real numbers a,b,c are in harmonic progression, then which of the following is true ?

a1​,b,c1​ are in AP

bc1​,ca1​,ab1​ are in HP

ab,bc,ca are in HP

b1​,c1​,a1​ are in HP