If $a_{r}$ is the coefficient of $x^{r - 1}$ in ${(1 + x)}^{n} + {(1 + x)}^{n + 1} + ... + {(1 + x)}^{n + k}\ (n

Question

If $a_{r}$ is the coefficient of $x^{r - 1}$ in ${(1 + x)}^{n} + {(1 + x)}^{n + 1} + ... + {(1 + x)}^{n + k}\ (n < r - 1 \leq n + k)$, then $\sum_{r = 0}^{n + k + 1}{{( - 1)}^{r}a_{r}}$ is equal to

Dr. Abubacker · Accepted Answer

(a)
Coefficient of $x^{r - 1}$ in
$(1 + x)^{n} + (1 + x)^{n + 1} + ... + (1 + x)^{n + k}$
$= \ ^{n}C_{r - 1} + \ ^{n + 1}C_{r - 1} + ... + \ ^{n + k}C_{r - 1}$
$= \ ^{n}C_{r} + \ ^{n}C_{r - 1} + \ ^{n + 1}C_{r - 1} + ... + \ ^{n + k}C_{r - 1} - \ ^{n}C_{r}$
$= \ ^{n + k + 1}C_{r} - \ ^{n}C_{r}$
Now, $\sum_{r = 0}^{n + k + 1}{{( - 1)}^{r}a_{r}} = \sum_{r = 0}^{n + k + 1}{( - 1)}^{r}\ ^{n + k - 1}C_{r} - \sum_{r = 0}^{n + k + 1}{( - 1)}^{r}\ ^{n}C_{r} = 0$

Q. If $a_{r}$ is the coefficient of $x^{r - 1}$ in ${(1 + x)}^{n} + {(1 + x)}^{n + 1} + ... + {(1 + x)}^{n + k}\ (n < r - 1 \leq n + k)$, then $\sum_{r = 0}^{n + k + 1}{{( - 1)}^{r}a_{r}}$ is equal to

Explanation

Reviewed by Dr. Abubacker

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