Home
Class 12
MATHS
If Cr=(n !)/([r !(n-r)]), the prove that...

If `C_r=(n !)/([r !(n-r)]),` the prove that `sqrt(C_1)+sqrt(C_2)++sqrt(C_n)

Text Solution

Verified by Experts

A.M. of `(1//2)th` powers `lt (1//2)th` power of A.M.
`therefore ((C_1)^((1)/(2))+(C_2)^((1)/(2))+...+(C_n)^((1)/(2)))/(n)lt ((C_1+C_2+....C_n)/(n))^(1//2)`
or ` (sqrt(C_1)+sqrt(C_2)+....+ sqrt(C_n))/(n) lt ((2^n-1)/(n))^(1//2)`
or ` sqrt(C_1)+sqrt(C_2)+....+sqrt(C_n)lt (n sqrt((2^n-1)))/(sqrt(n))`
Hence,
` sqrt(C_1)+sqrt(C_2)+....+sqrt(C_n) lt sqrt([n(2^n-1)])`
Promotional Banner

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise EXERCISES (Single Correct answer type)|20 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Multiple correct answers type|5 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Concept Application Eexercises 6.3|6 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE PUBLICATION|Exercise Single correct Answer|21 Videos

  • INTEGRALS

    CENGAGE PUBLICATION|Exercise All Questions|762 Videos

Similar Questions

Explore conceptually related problems

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)

If (1+x)^n=sum_(r=0)^n C_r x^r , then prove that C_1+2C_2+3C_3+....+n C_n=n2^(n-1)dot .

If n=12 m(m in N), prove that .^n C_0-(.^n C_2)/((2+sqrt(3))^2)+(.^n C_4)/((2+sqrt(3))^4)-(.^n C_6)/((2+sqrt(3))^6)+.......= (-1)^m((2sqrt(2))/(1+sqrt(3)))^ndot

In triangle A B C ,l e tR=c i r c u m r a d i u s ,r= i n r a d i u sdot If r is the distance between the circumcenter and the incenter, the ratio R/r is equal to (a) sqrt(2)-1 (b) sqrt(3)-1 (c) sqrt(2)+1 (d) sqrt(3)+1

Sum of 1/(sqrt(2)+sqrt(5))+1/(sqrt(5)+sqrt(8))+1/(sqrt(8)+sqrt(11))+1/(sqrt(11)+sqrt(14))+..to n terms= (A) n/(sqrt(3n+2)-sqrt(2)) (B) 1/3 (sqrt(2)-sqrt(3n+2) (C) n/(sqrt(3n+2)+sqrt(2)) (D) none of these

(n)p_(r)=k^(n)C_(n-r),k=

If .^(2n)C_(r)=.^(2n)C_(r+2) find the value of r.

If nC_r + nC_(r+1) = (n+1)C_x then x=?

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

The value of ("lim")_(nvecoo)sum_(r=1)^(4n)(sqrt(n))/(sqrt(r)(3sqrt(r)+sqrt(n))^2) is equal to 1/(35) (b) 1/4 (c) 1/(10) (d) 1/5