QuizSolver
  • Bank PO
  • CBSE
  • IIT JEE
 
 

Solved Subjective Question on Binomial Expression Set 4

Posted on - 04-01-2017

Math

IIT JEE

Example.1

Show that C0 –22C1 + 32C2 - . . . +(–1)n(n+1)2Cn = 0, n> 2.

Solution

We have

C0 + C1x + C2x2 + . . . + Cnxn = (1+x)n . . . (1).

Multiplying (1) by x

C0x + C1x2+C2x3 +. . . +Cnxn+1 = x(1+x)n . . . (2).

Differentiating (2) w.r.t. x, we have .

C0 + 2c1x +3C2x2 + . . . +(n+1) Cnxn = (1+x)n + nx(1+x)n–1 .

= [(n+1)x+1] (1+x)n–1 . . . (3).

Multiplying (3) by x, we get

C0x + 2C1x2 + 3C2x3 + . . . + (n+1) Cnxn–1 = [(n+1)x2+x](1+x)n-1 . . . (4).

Differentiating (4) w.r.t x , we get .

C0 + 22 C1x + 32 C2x2 + . . . + (n+1)2 Cnxn.

= [2(n+1)x+1)] (1+x)n–1 + (n–1) [(n+1)x2 + x] ´ (1+x)n–2 . . . (5).

Putting x = –1 in (5) , we get

C0–22C1 + 32C2 - . . . +(–1)n (n+1)2 Cn = 0.

Example.2

Prove that C0Cr + C1Cr+1 + C2Cr+2 + . . . + Cn–r Cn =

Solution

We have,

C0 + C1x + C2x2 + . . . + Cnxn = (1+x)n . . . (1).

Also, C0xn + C1xn–1+ C2xn–2 + . . . + Cn = (x+1)n . . . (2).

Multiplying (1) and (2), we get

(C0+C1x+C2x2 +…+ Cnxn) (C0xn+ C1xn–1 +C2xn–2 + …+Cn)

= (1+x) 2n . . . (3).

Equating coefficinet of xn–r from both sides of (3), we get

C0Cr + C1Cr+1 + C2Cr+2 + ------------+ Cn–rCn = 2n Cn–r =

Example.3

Find the sum of the series

Solution

The given series

Now, =

Similarly, etc.

Hence, the gives series =

= =

Example.4

Let k and n ∈ I+ and Sk = 1k + 2k +. . . . + nk, show that , m+1C1S1 + m+2C2S2+ . . . . + m+1CmSm = (n+1)m+1 – (n+1) . .

Solution

We have


&⇒

Putting p = 1, 2, 3,…,n and adding, we get


&⇒

Example.5

Show that .

Solution

We have, for r ≥ 0,

= =

Thus, =

= [put r+2 = s]

=

=

= =

But 2n+3 + (-1)n =

Thus,

 
Quadratic Equations - Solved Objective Questions Part 2 for Conceptual Clarity
Quadratic Equations - Solved Objective Questions Part 1 for Conceptual Clarity
Solved Objective Question on Probability Set 2
Solved Objective Question on Probability Set 1
Solved Objective Question on Progression and Series Set 2
Solved Objective Question on Permutations and Combinations Set 3
Solved Objective Question on Permutations and Combinations Set 2
Solved Objective Question on Progression and Series Set 1
Solved Objective Question on Permutations and Combinations Set 1
Quadratic Equations - Solved Subjective Questions Part 4
Quadratic Equations - Solved Subjective Questions Part 2
Quadratic Equations - Solved Subjective Questions Part 3
Quadratic Equations - Solved Subjective Questions Part 1
Solved Subjective Questions on Circle Set 9
Solving Equations Reducible to Quadratic Equations
Theory of Polynomial Equations and Remainder Theorem
Solved Subjective Questions on Circle Set 8
Solving Quadratic Inequalities Using Wavy Curve Methods
Division and Distribution of Objects - Permutation and Combination
Basics of Quadratic Inequality or Inequations

Comments