Home
Class 12
MATHS
If n >2, then prove that C1(a-1)-C2xx(a-...

If `n >2,` then prove that `C_1(a-1)-C_2xx(a-2)++(-1)^(n-1)C_n(a-n)=a ,w h e r eC_r=^n C_rdot`

Text Solution

Verified by Experts

`S = C_(1)(a-1)- C_(2)(a-2) + "…." + (-1)^(n-1)C_(n)(a-n)`
`:. T_(r) = (-1)^(r-1)(a-r).^(n)C_(r)`
`= (-1)^(r-1)(a.^(n)C_(r) - r.^(n)C_(r))`
`= (-1)^(r-1)(a.^(n)C_(r)-n.^(n-1)C_(r-1))`
` = - a (-1)^(r ). .^(n)C_(r) - n (-1)^(r=1 xx n - 1) C_(r-1)`
Now, `S = underset(r=1)overset(n)sumT_(r)`
`= -a[(1-1)^(n)-.^(n)C_(0)] - n(1-1)^(n-1)`
`= an`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Example|10 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 8.1|17 Videos

  • AREA

    CENGAGE PUBLICATION|Exercise Comprehension Type|2 Videos

  • CIRCLE

    CENGAGE PUBLICATION|Exercise For problems 3 and 4|2 Videos

Similar Questions

Explore conceptually related problems

.^(n-1)C_(r)+^(n-1)C_(r-1)=

show that ^nC_r+ ^(n-1)C_(r-1)+ ^(n-1)C_(r-2)= ^(n+1)C_r

Prove that "^n C_r+^(n-1)C_r+...+^r C_r=^(n+1)C_(r+1) .

Prove that sum_(r=0)^(2n)(r. ^(2n)C_r)^2=n^(4n)C_(2n) .

Find the sum .^1C_0+^2C_1+^3C_2+....+^(n+1)C_n ,where C_r=^n C_rdot

Prove that , .^(n)C_(r)+3.^(n)C_(r-1)+3.^(n)C_(r-2)+^(n)C_(r-3)=^(n+3)C_(r)

Show that , .^(n)C_(r)=(n-r+1)/(r).^(n)C_(r-1) .

Prove that (C_1)/2+(C_3)/4+(C_5)/6+=(2^(n)-1)/(n+1)dot

Show that .^(n)C_(r)+.^(n-1)C_(r-1)+.^(n-1)C_(r-2)=.^(n+1)C_(r) .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0) *""^(2n)C_(n) - C_(1) *""^(2n-2)C_(n) + C_(2) *""^(2n-4) C_(n) -…= 2^(n)