Question

If \[a,b,c\] are in arithmetic progression, then the value of \[(a + 2b - c)(2b + c - a)(a + 2b + c)\] is
A. \[16abc\]
B. \[4abc\]
C. \[8abc\]
D. \[3abc\]

Answer 1

Hint: A sequence of numbers is called an arithmetic progression if the difference between any two consecutive terms is always the same. In simple terms, it means that the next number in the series is calculated by adding a fixed number to the previous number in the series. This fixed number is called the common difference.

Complete step by step answer:
An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.
The fixed number that must be added to any term of an AP to get the next term is known as the common difference of the AP.
If \[a\] is the first term and \[d\] is the common difference then the nth term of an AP is \[a + (n - 1)d\] .
The finite portion of an AP is known as finite AP and therefore the sum of finite AP is known as arithmetic series. The behaviour of the sequence depends on the value of a common difference.
If the value of “d” is positive, then the member terms will grow towards positive infinity
If the value of “d” is negative, then the member terms grow towards negative infinity
We are given that \[a,b,c\] are in arithmetic progression.
Therefore by property of an arithmetic progression we know \[2b = a + c\]
Consider \[(a + 2b - c)(2b + c - a)(a + 2b + c)\]
Puting the value of \[2b\] from the above equation we get \[(a + a + c - c)(a + c + c - a)(a + a + c + c)\]
Which on simplification becomes \[(2a)(2c)(2a + 2c)\]
                                                            \[ = 8ac(a + c)\]
Hence now putting \[a + c = 2b\]
We get \[(a + 2b - c)(2b + c - a)(a + 2b + c) = 16abc\]
So, the correct answer is “Option A”.

Note: A sequence of numbers is called an arithmetic progression if the difference between any two consecutive terms is always the same. The fixed number that must be added to any term of an AP to get the next term is known as the common difference of the AP.