Question

Are \(a,b\) and \(c\) in arithmetic progression? I) \(5a,5b\) and \(5c\) are in arithmetic progression.
II) \(2a,3b\) and \(4c\) are in arithmetic progression.

• A. Statement I alone is sufficient to answer the question
• B. Statement II alone is sufficient to answer the question
• C. Both the statements I and II are sufficient to answer the question but neither statement alone is not sufficient
• D. Both the statements I and II together are not sufficient to answer the question and additional data is required

Hint:

If \(ka,kb,kc\) are in AP for non-zero \(k\), then \(a,b,c\) are also in AP.

Answer 1

The Correct Option is A

Concept:
Three numbers \(a,b,c\) are in arithmetic progression if \[ 2b=a+c \] Multiplying all terms of an arithmetic progression by the same non-zero number does not change the AP property.

Step 1: Check Statement I. Statement I says \[ 5a,\ 5b,\ 5c \] are in arithmetic progression. So, \[ 2(5b)=5a+5c \] \[ 10b=5a+5c \] Divide by \(5\): \[ 2b=a+c \] Therefore, \[ a,b,c \] are in arithmetic progression. Thus Statement I alone is sufficient.

Step 2: Check Statement II. Statement II says \[ 2a,\ 3b,\ 4c \] are in arithmetic progression. So, \[ 2(3b)=2a+4c \] \[ 6b=2a+4c \] \[ 3b=a+2c \] This does not necessarily imply \[ 2b=a+c \] Therefore, Statement II alone is not sufficient.

Step 3: Final answer. \[ \boxed{\text{Statement I alone is sufficient}} \]