If 1a, 1b, 1c are in AP, which one of the following is in AP

It is known that, if we multiply or divide the numbers which are in AP with a particular number, then still that number will be in AP.Now, multiply a + b + c to all the three numbers.So, the numbers will become, a+b+ca, a+b+cb, a+b+cc.If we subtract or add some particular number from the numbers which are in AP, still these numbers will be in AP.Now, subtract 1 from all these three numbers.a+b+ca – 1, a+b+cb – 1, a+b+cc – 1a+b+c-aa, a+b+c-bb, a+b+c-ccb+ca, a+cb, a+bc are in AP.Hence, option 3 is correct.

If 1a, 1b, 1c are in AP, which one of the following is in AP

It is known that, if we multiply or divide the numbers which are in AP with a particular number, then still that number will be in AP.Now, multiply a + b + c to all the three numbers.So, the numbers will become, a+b+ca, a+b+cb, a+b+cc.If we subtract or add some particular number from the numbers which are in AP, still these numbers will be in AP.Now, subtract 1 from all these three numbers.a+b+ca – 1, a+b+cb – 1, a+b+cc – 1a+b+c-aa, a+b+c-bb, a+b+c-ccb+ca, a+cb, a+bc are in AP.Hence, option 3 is correct.

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If $\frac{1}{a}$ , $\frac{1}{b}$ , $\frac{1}{c}$ are in AP, which one of the following is in AP

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\frac{b - c}{a}

\frac{b + c}{a}

\frac{b - c}{b}

\frac{c - a}{b}

\frac{b + c}{a}

\frac{b - c}{b}

\frac{a + b}{c}, \frac{b + c}{a}

\frac{a + c}{b}

\frac{2 a + 2 b}{c}

\frac{2 a - 2 b}{c}

\frac{2 c - 2 b}{c}

answer is C.

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Detailed Solution

It is known that, if we multiply or divide the numbers which are in AP with a particular number, then still that number will be in AP.
Now, multiply a + b + c to all the three numbers.
So, the numbers will become,

\frac{a + b + c}{a}

\frac{a + b + c}{b}

\frac{a + b + c}{c}

.
If we subtract or add some particular number from the numbers which are in AP, still these numbers will be in AP.
Now, subtract 1 from all these three numbers.