Given that [tex]\(a, b, c, d,\)[/tex] and [tex]\(e\)[/tex] form an arithmetic progression (A.P.), we can denote the common difference by [tex]\(d\)[/tex]. In an A.P., each term after the first term is obtained by adding the common difference to the previous term. Therefore, we can express the terms [tex]\(b\)[/tex], [tex]\(c\)[/tex], [tex]\(d\)[/tex], and [tex]\(e\)[/tex] in terms of [tex]\(a\)[/tex] and [tex]\(d\)[/tex] as follows:
- [tex]\(b = a + d\)[/tex]
- [tex]\(c = a + 2d\)[/tex]
- [tex]\(d = a + 3d\)[/tex]
- [tex]\(e = a + 4d\)[/tex]
Now, we need to find the value of the expression [tex]\(a - 4b + 6c - 4d + e\)[/tex]. Substituting the values of [tex]\(b\)[/tex], [tex]\(c\)[/tex], [tex]\(d\)[/tex], and [tex]\(e\)[/tex] in terms of [tex]\(a\)[/tex] and [tex]\(d\)[/tex] into the expression, we get:
[tex]\[
a - 4(a + d) + 6(a + 2d) - 4(a + 3d) + (a + 4d)
\][/tex]
Let's simplify this step by step:
1. Substitute the expressions for [tex]\(b\)[/tex], [tex]\(c\)[/tex], [tex]\(d\)[/tex], and [tex]\(e\)[/tex]:
[tex]\[
a - 4(a + d) + 6(a + 2d) - 4(a + 3d) + (a + 4d)
\][/tex]
2. Distribute the coefficients:
[tex]\[
a - 4a - 4d + 6a + 12d - 4a - 12d + a + 4d
\][/tex]
3. Combine like terms:
[tex]\[
(a - 4a + 6a - 4a + a) + (-4d + 12d - 12d + 4d)
\][/tex]
4. Simplify the coefficients:
[tex]\[
(1 - 4 + 6 - 4 + 1)a + (-4 + 12 - 12 + 4)d
\][/tex]
5. Calculate the sum of the constants:
[tex]\[
0a + 0d
\][/tex]
This results in the simplified expression:
[tex]\[
4a + 8d
\][/tex]
Thus, the value of the original expression [tex]\(a - 4b + 6c - 4d + e\)[/tex] simplifies to:
[tex]\[
4a + 8d
\][/tex]
Therefore, the value of [tex]\(a - 4b + 6c - 4d + e\)[/tex] is [tex]\(4a + 8d\)[/tex].
