If n- 2 then prove that C1a-1-C2-a-2-1n-1Cna-n-a-where Cr- n C r

(1 x)^n=C_0+C_1( x)+C_2( x)^2+.......+C_n( x)^n.........(1) Differentiating on both sides #160; n(1 x)^n 1=0 C_1+2C_2(x) 3C_3(x)^2+......+n( 1)^n(x) ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Sum of Coefficients of All Terms

If n> 2 the...

Question

If $n > 2$ then prove that $C_{1} (a - 1) - C_{2} \times (a - 2) + . . . . . + {(- 1)}^{n - 1} C_{n} (a - n) = a$ ,where $C_{r} = {^{n} C}_{r}$

Open in App

Solution

$(1 - x)^{n} = C_{0} + C_{1} (- x) + C_{2} (- x)^{2} + . . . . . . . + C_{n} (- x)^{n} . . . . . . . . . (1)$
Differentiating on both sides
$- n (1 - x)^{n - 1} = 0 - C_{1} + 2 C_{2} (x) - 3 C_{3} (x)^{2} + . . . . . . + n (- 1)^{n} (x)^{n - 1} . . . . . . (2)$
When $x = 1$
$(1)$ becomes $0 = C_{0} - C_{1} + C_{2} - . . . . . . . . + (- 1)^{n - 1} C_{n}$
$⟹ C_{1} - C_{2} + C_{3} - . . . . . + (- 1)^{n - 1} C_{n} = C_{0} = 1$
$(2)$ becomes $0 = - C_{1} + 2 C_{2} - 3 C_{3} + . . . . + (- 1)^{n} n C_{n}$
$L H S = C_{1} (a - 1) - C_{2} (a - 2) + . . . . . . . + (- 1)^{n - 1} C_{n} (a - n)$
$= a (C_{1} - C_{2} + C_{3} - . . . . + (- 1)^{n - 1} C_{n}) + (- C_{1} + 2 C_{2} - 3 C_{3} + . . . . + (- 1)^{n} n C_{n}) = a (1) + 0 = a = R H S$
$L H S = R H S$
Hence Proved

Suggest Corrections

Similar questions

Q. If

n > 2

, then find the value of

C_{1} {(a - 1)}^{2} - C_{2} {(a - 2)}^{2} + C_{3} {(a - 3)}^{2} - . . . . . + {(- 1)}^{n - 1} C_{n} {(a - n)}^{2}

where

C_{r}

stands for

{^{n} C}_{r}

Q. If

n

is a positive integer, prove that

n \sum r = 1 r^{3} {(\frac{^{n} C_{r}}{^{n} C_{r - 1}})}^{2} = \frac{n {(n + 1)}^{2} (n + 2)}{12}

Q. If

1 \leq r \leq n

, then prove that

{^{n} C}_{r} + {^{n} C}_{r - 1} = {^{n + 1} C}_{r}

^{m} C_{r} +^{m} C_{r - 1} .^{n} C_{1} +^{m} C_{r - 2},^{n} C_{2} + . . . +^{m} C_{1} .^{n} C_{r - 1} +^{n} C_{r} =

Q. If

c_{0}, c_{1}, c_{2}, . . . . . . . c_{n}

denote the coefficients in the expansion of

{(1 + x)}^{n}

, prove that

c_{0} c_{r} + c_{1} c_{r + 1} + c_{2} c_{r + 2} + . . . . + c_{n - r} c_{n} = \frac{| 2 n - - -}{| n - r - ---- - | n + r - ---- -}

Join BYJU'S Learning Program

If n>2 then prove that C1(a−1)−C2×(a−2)+.....+(−1)n−1Cn(a−n)=a,where Cr=nCr

If $n > 2$ then prove that $C_{1} (a - 1) - C_{2} \times (a - 2) + . . . . . + {(- 1)}^{n - 1} C_{n} (a - n) = a$ ,where $C_{r} = {^{n} C}_{r}$