Question

If the numbers $a,b,c,d,e$ form an A.P, then the value of $a - 4b + 6c - 4d + e$ is-
$\eqalign{
  & 1)1 \cr
  & 2)2 \cr
  & 3)0 \cr
  & 4)3 \cr} $

Answer 1

Hint: It is given that the numbers are in an Arithmetic Progression. That means, they will have a common difference between each term. We can express the given numbers in terms of the common difference. Then by substituting those terms in the given equation, we can find out the final answer.
Formula used in the question is:
For Arithmetic Progression:
${a_n} = {a_1} + \left( {n - 1} \right)d$
Where,
${a_n}$ is the ${n^{th}}$term in the sequence.
${a_1}$ is the ${1^{st}}$term in the sequence.
$d$ is the common difference between the terms.

Complete step-by-step solution:
The given numbers are $a,b,c,d,e$ and they are in an Arithmetic Progression.
Let $D$ be the common difference of the given AP.
Now let us express $a,b,c,d,e$ in terms of the common difference $D$ using the above equation.
$\eqalign{
  & \Rightarrow b = a + D \cr
  & \Rightarrow c = a + 2D \cr
  & \Rightarrow d = a + 3D \cr
  & \Rightarrow e = a + 4D \cr} $
Now, we can substitute these values for the given equation.
$$a - 4b + 6c - 4d + e = a - 4(a + D) + 6(a + 2D) - 4(a + 3D) + a + 4D$$
By simplifying the LHS, we get,
$\eqalign{
  & = a - 4a - 4D + 6a + 12D - 4a - 12D + a + 4D \cr
  & = 8a - 8a \cr
  & = 0 \cr} $
Therefore, the final is $0$
Hence, option (3) is the correct answer.

Additional Information:
An AP is a sequence of numbers where the difference in the consecutive numbers is constant.

Note: An AP is a sequence of numbers where the difference in the consecutive numbers is constant. The key term is that the given terms are in AP. Therefore, use the given formula of AP. Since it is already used in the question, use D as the common difference in the solution to avoid any confusions. Be careful while solving the equation since there are more terms. Keep all the variables together, then it will be easier to cancel out and get to the final answer.