Explanation

(c)
We can write,
${aC}_{0} - (a + d)C_{1} + (a + 2d)C_{2} - ...$ upto $(n + 1)$ terms
$= a\left( C_{0} - C_{1} + C_{2} - ... \right) + d({- C}_{1} + {2C}_{2} - {3C}_{3} + ...)$ …(i)
We know,
${(1 - x)}^{n} = C_{0} - C_{1}x + C_{2}x^{2} - ... + {( - 1)}^{n}C_{n}x^{n}$ …(ii)
On differentiating Eq. (ii) w.r.t. $x$, we get
$- n{(1 - x)}^{n - 1} = {- C}_{1} + {2C}_{2}x - ...{( - 1)}^{n}C_{n}{nx}^{n - 1}$ …(iii)
On putting $x = 1$ in Eqs. (ii) and (iii), we get
$C_{0} - C_{1} + C_{2} - ... + {( - 1)}^{n}C_{n} = 0$ …(iv)
and ${- C}_{1} + {2C}_{2} - ... + {( - 1)}^{n}{nC}_{n} = 0$ …(v)
From Eq. (i),
${aC}_{0} - (a + d)C_{1} + (a + 2d)C_{2} - ...$ upto $(n + 1)$ terms
$= a \bullet 0 + d \bullet 0 = 0$ [from Eqs. (iv) and (v)]