Math Problem Statement
Assertion: Let the positive numbers π , π , π a,b,c be in Arithmetic Progression (A.P.). Then 1 π π , 1 π π , 1 π π bc 1 β , ac 1 β , ab 1 β are also in A.P.
Reason: If each term of an A.P. is divided by π π π abc, the resulting sequence is also in A.P.
Solution
Let's analyze the assertion and reason step-by-step.
Assertion:
We are given that , , and are positive numbers in Arithmetic Progression (A.P.). The claim is that the terms:
are also in A.P..
Reason:
We are asked to reason that if each term of an A.P. is divided by the product , the resulting sequence will also be in A.P..
Step-by-Step Solution:
1. Definition of Arithmetic Progression:
A sequence is in A.P. if the common difference between consecutive terms is constant, i.e.,
2. Given A.P.: are in A.P..
For , , and to be in A.P., the common difference condition must hold. Hence,
3. Sequence to Verify: .
Let us define these terms as , , and .
We need to verify if the sequence satisfies the condition for an A.P.:
Letβs compute the differences:
Now, since (from the A.P. condition for ), we have:
Thus,
Therefore, the sequence is indeed in A.P..
Conclusion:
The given assertion is correct. The sequence is in A.P. when are in A.P..
Would you like more details on any part of the solution or have additional questions on the topic? Here are some related questions you may find useful:
- What is the general form of an A.P. and how is the common difference calculated?
- Can you prove a sequence of fractions is in A.P. if their numerators are in A.P.?
- How does the harmonic progression relate to A.P.?
- What other types of progressions (like geometric progression) have similar properties?
- How can we generalize this result to more than three terms in A.P.?
Tip: In problems involving A.P., always check the condition for three terms to confirm whether they follow an A.P.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Arithmetic Progression (A.P.)
Sequences
Fractions
Formulas
x2 - x1 = x3 - x2 for A.P.
A.P. condition: 2b = a + c
Theorems
Properties of Arithmetic Progression
Suitable Grade Level
Grades 10-12
Related Recommendation
Proving Arithmetic Progression in Terms of Reciprocals and Squares
Proof of Symmetric Rational Expression Given a = b + c, b = c + a, c = a + b
Prove that a, c, e are in G.P. when a, b, c are in A.P., b, c, d are in G.P., and c, d, e are in H.P.
Calculate Value of b/a in Arithmetic Progression (A.P.) Problem
Proof: If ac divides bc, then a divides b