# If (a – b), (b – c), (c – a) are in GP then prove that

Question:

If $(a-b),(b-c),(c-a)$ are in GP then prove that $(a+b+c)^{2}=3(a b+b c+$ ca).

Solution:

To prove: $(a+b+c)^{2}=3(a b+b c+c a)$.

Given: $(a-b),(b-c),(c-a)$ are in GP

Formula used: When $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{GP}, \mathrm{b}^{2}=\mathrm{ac}$

As, $(a-b),(b-c),(c-a)$ are in GP

$\Rightarrow(b-c)^{2}=(a-b)(c-a)$

$\Rightarrow b^{2}-2 c b+c^{2}=a c-a^{2}-b c+a b$

$\Rightarrow a^{2}+b^{2}+c^{2}-b c-a c-a b=0$

Adding 3(ab + bc + ac) both side

$\Rightarrow a^{2}+b^{2}+c^{2}-b c-a c-a b+3(a b+b c+a c)=3(a b+b c+a c)$

$\Rightarrow a^{2}+b^{2}+c^{2}+2 b c+2 a c+2 a b=3(a b+b c+a c)$

$\Rightarrow(a+b+c)^{2}=3(a b+b c+a c)$

Hence Proved
The text passage contains a mathematical question that requires a proof, and the solution to that question is provided in the text.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# If (a – b), (b – c), (c – a) are in GP then prove that

Question:

If $(a-b),(b-c),(c-a)$ are in GP then prove that $(a+b+c)^{2}=3(a b+b c+$ ca).

Solution:

To prove: $(a+b+c)^{2}=3(a b+b c+c a)$.

Given: $(a-b),(b-c),(c-a)$ are in GP

Formula used: When $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{GP}, \mathrm{b}^{2}=\mathrm{ac}$

As, $(a-b),(b-c),(c-a)$ are in GP

$\Rightarrow(b-c)^{2}=(a-b)(c-a)$

$\Rightarrow b^{2}-2 c b+c^{2}=a c-a^{2}-b c+a b$

$\Rightarrow a^{2}+b^{2}+c^{2}-b c-a c-a b=0$

Adding 3(ab + bc + ac) both side

$\Rightarrow a^{2}+b^{2}+c^{2}-b c-a c-a b+3(a b+b c+a c)=3(a b+b c+a c)$

$\Rightarrow a^{2}+b^{2}+c^{2}+2 b c+2 a c+2 a b=3(a b+b c+a c)$

$\Rightarrow(a+b+c)^{2}=3(a b+b c+a c)$

Hence Proved
user
The text passage contains a mathematical question that requires a proof, and the solution to that question is provided in the text.

TYPE: SINGLE_QUESTION_WITH_ANSWER