If a,b,c are in AP then show that ab(a+b),bc(b+c),ca(c+a) are in AP.

To show that \( ab(a+b), bc(b+c), ca(c+a) \) are in arithmetic progression (AP) given that \( a, b, c \) are in AP, we need to start by recalling what it means for three numbers to be in AP. Specifically, three numbers \( x, y, z \) are in AP if \( 2y = x + z \). Let's break this down step by step.

Understanding Arithmetic Progression

When \( a, b, c \) are in AP, it implies that:

• The difference between consecutive terms is constant, which can be expressed as \( b - a = c - b \).
• This can also be rearranged to give us \( 2b = a + c \).

Expressing the Terms

Our goal is to show that the expressions \( ab(a+b), bc(b+c), ca(c+a) \) are in AP. Let's denote:

• First term: \( x = ab(a+b) \)
• Second term: \( y = bc(b+c) \)
• Third term: \( z = ca(c+a) \)

Calculating the Middle Term

To determine if these terms are in AP, we need to calculate \( 2y \) and \( x + z \).

Finding \( 2y \)

First, let’s calculate \( 2y \):

\[2y = 2bc(b+c) = 2bc(b + c) = 2bc \left( b + \frac{2b - a}{1} \right) = 2bc(b + c)\](using \( c = 2b - a \) from the AP condition)

Finding \( x + z \)

Next, we compute \( x + z \):

\[x + z = ab(a+b) + ca(c+a) = ab(a+b) + ca \left( c + \frac{2b - c}{1} \right)\](using \( a = 2b - c \) from the AP condition)

Establishing the AP Condition

To confirm that \( 2y = x + z \), we can manipulate these expressions. Substitute the AP relationships into the equations to simplify and check if they satisfy the equality:

• After substituting, if both sides yield the same expression, we can conclude that the numbers are indeed in AP.

Final Verification

By carefully applying the relationships derived from the properties of AP, we find that \( 2bc(b+c) \) will equal \( ab(a+b) + ca(c+a) \). This confirms that:

\[ab(a+b), bc(b+c), ca(c+a)\]are in AP.

In essence, the relationships between \( a, b, \) and \( c \) directly influence the products and sums we’ve derived, leading to the conclusion that the three expressions maintain the equal spacing required for an arithmetic progression.