Question

If the numbers $a, b, c, d, e$ form an A.P with $a = 1$, then find the value of $a - 4b + 6c - 4d + e$.
A. 1
B. 2
C. 0
D. 3

Answer 1

Hint: The given variables are in an arithmetic progression so there will be a common difference and nth term can be calculated using suitable formulas.

Formula Used:
The general formula of arithmetic progression of nth term is ${t_n} = a + (n - 1)d$, where $a$ is the first term, $n$ is the term number and $d$ is the common difference.

Complete step by step solution:
Given that $a, b, c, d, e$ are in the arithmetic progression.
So, the first term of the arithmetic progression is a
${t_1} = a = 1$
Assuming the common difference is $x$ . Then, using the nth term formula, rewrite each term as follows:
${t_n} = a + (n - 1)d$
The value of second term is
${t_2} = 1 + (2 - 1)x$
$ \Rightarrow b = 1 + x$
The value of third term is
${t_3} = 1 + (3 - 1)x$
$ \Rightarrow c = 1 + 2x$
The value of fourth term is
${t_4} = 1 + (4 - 1)x$
$ \Rightarrow d = 1 + 3x$
The value of fifth term is
${t_5} = 1 + (5 - 1)x$
$ \Rightarrow e = 1 + 4x$
Next putting the above values in the given expression $a - 4b + 6c - 4d + e$ , we get
$a - 4b + 6c - 4d + e$
Substitute the values of $a, b, c, d, e$
$ = 1 - 4(1 + x) + 6(1 + 2x) - 4(1 + 3x) + 1 + 4x$
Simplifying this expression, we get
$ = 1 - 4 - 4x + 6 + 12x - 4 - 12x + 1 + 4x$
Simplifying this expression, we get
$ = 6 - 6 = 0$

Option ‘C’ is correct

Note: Remember the properties and related formulas for an Arithmetic Progression. Also, we need to take care of multiplication of signs.
$\left( + \right) \times \left( - \right) = \left( - \right)$
$\left( - \right) \times \left( - \right) = \left( + \right)$
$\left( - \right) \times \left( + \right) = \left( - \right)$