Home
Class 12
MATHS
Find the sum .^(n)C(1) + 2 xx .^(n)C(2)...

Find the sum `.^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "……" + n xx .^(n)C_(n)`.

Text Solution

Verified by Experts

Method I :
`.^(n)C_(0)+2xx.^(n)C_(1)+3xx.^(n)C_(2)+"...."+(n+1)xx .^(n)C_(n)`
`= underset(r=0)overset(n)sum(r+1).^(n)C_(r)`
`=underset(r=0)overset(n)sum[r.^(n)C_(r)+.^(n)C_(r)]`
`=n underset(r=0)overset(n)sum.^(n-1)C_(r-1)+underset(r=0)overset(n)sum.^(n)C_(r)`
`= n(.^(n-1)C_(0) + .^(n-1)C_(1) + .^(n-1)C_(2)+"..."+.^(n-1)C_(n-1)) + (.^(n)C_(0)+.^(n)C_(1)+.^(n)C_(2)+"....."+.^(n)C_(n))`
`= n2^(n-1) + 2^(n)`
`= (n+2)2^(n-1)`
Method II :
We have `(1+x)^(n) = .^(n)C_(0)+.^(n)C_(1)x+.^(n)C_(2)x^(2)+"....."+.^(n)C_(n)x^(n)`
`:. x(1+x)^(n) = .^(n)C_(0)x+.^(n)C_(1)x^(2)+.^(n)C_(2)x^(3) + "....." + .^(n)C_(n)x^(n+1)`
Differentiating w.r.t. x, we get
`n(n1+x)^(n-1)x+(1+x)^(n)=.^(n)C_(0)+2xx.^(n)C_(1)x+ 3xx.^(n)C_(2)x^(2)+"..."+(n+1)xx.^(n)C_(n)x^(n)`
Putting `x = 1`, we get
`n2^(n-1)+2^(n)=.^(n)C_(0) +2xx.^(n)C_(1)+3xx.^(n)C_(2)+"....."+(n+1)xx.^(n)C_(n)`
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

Prove that .^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "…." + n xx .^(n)C_(n) = n2^(n-1) . Hence, prove that .^(n)C_(1).(.^(n)C_(2))^(2).(.^(n)C_(3))^(3)"......."(.^(n)C_(n))^(n) le ((2^(n))/(n+1))^(.^(n+1)C_(2)) AA n in N .

Find the sum 1 xx 2 xx .^(n)C_(1) + 2 xx 3 xx .^(n)C_(2) + "….." + n xx (n+1) xx .^(n)C_(n) .

Prove that .^(n)C_(0) +5 xx .^(n)C_(1) + 9 xx .^(n)C_(2) + "…." + (4n+1) xx .^(n)C_(n) = (2n+1) 2^(n) .

If .^(n)C_(3)=.^(n)C_(2) , then find .^(n)C_(2) .

If .^(n)C_(4)=.^(n)C_(2) , then find .^(n)C_(3) .

If .^(n)C_(7)=.^(n)C_(2) , then find .^(n)C_(2) .

Prove that .^(n)C_(0) - .^(n)C_(1) + .^(n)C_(2) - .^(n)C_(3) + "……" + (-1)^(r) .^(n)C_(r) + "……" = (-1)^(r ) xx .^(n-1)C_(r ) .

If .^(n)C_(5)=.^(n)C_(3) , then find .^(n)C_(4) .

If .^(n)C_(8)=.^(n)C_(6) , then find .^(n)C_(2) .

Prove that .^(n)C_(0) + (.^(n)C_(1))/(2) + (.^(n)C_(2))/(3) + "……" +(. ^(n)C_(n))/(n+1) = (2^(n+1)-1)/(n+1) .