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State if the following is true or false

$b + c − a a, c + a − b b, a + b − c c are in A .P ., 1 a, 1 b, 1 c are in A .P ., a + b + c ≠ 0$ .

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True

False

answer is A.

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Detailed Solution

Given that

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

are in A.P..
According to the question,

b + c − a a + 2

c + a − b b + 2

a + b − c c + 2

are in A.P.
The series is in A.P since adding a constant term to all terms does not affect the progression.
So,

b + c − a + 2 a a

c + a − b + 2 b b

a + b − c + 2 c c

are in A.P,
Therefore,

b + c + a a

c + a + b b

a + b + c c

are in A.P.
Now,
Take the common term outside from

b + c + a a

c + a + b b

a + b + c c

.
The series

b + c + a a, c + a + b b, a + b + c c

becomes

a + b + c 1 a, 1 b, 1 c

Now, divide all the terms by a+b+c, we get,

\frac{1}{a}, \frac{1}{b}, \frac{1}{c}

So,

1 a

1 b

1 c

are in A.P because dividing all the terms by a common factor does not affect the progression.
It is proved that if

b + c − a a

c + a − b b

a + b − c c

are in A.P, then

1 a

1 b

1 c

are in A.P, provided

a + b + c ≠ 0

.
Therefore, the statement is true.
Hence, option 1 is correct.

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