For n , p ∈ N - { 1 } , the coefficient of x 3 in ( 1 - x ) - 1 / p ( 1 - x ) n =

( n p + 1 ) ( n p + p + 1 ) ( n p + 2 p + 1 ) p 3 × 3 !

( n p + 1 ) ( n p + p ) ( n p + 2 p ) 3 ! p 3

( n p + p ) ( n p + 2 p ) ( n p + 3 p ) 3 ! p 3

( n p + 1 ) ( n p + 2 ) ( n p + 3 ) 3 ! p 3

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The first negative term in the expansion of ${(1 + x)}^{27 / 5}$ $(0 < x < 1)$ is

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Important Questions on Binomial Theorem

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Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

In the expansion of

{(1 + \frac{3 x}{2})}^{- 5}

the coefficient of

x^{10}

is equal to the coefficient of

x^{10}

{(1 + a x)}^{n}, n \in N

then

n a =

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

The coefficient of

t^{4}

in the expansion of

{(\frac{1 - t^{6}}{1 - t})}^{3}

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

b

is very small as compared to the value of

a

, so that the cube and other higher powers of

\frac{b}{a}

can be neglected in the identity

$\frac{1}{a - b} + \frac{1}{a - 2 b} + \frac{1}{a - 3 b} + \dots . + \frac{1}{a - n b} = α n + β n^{2} + γ n^{3}$

then the value of $γ$ is :

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Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

For

n, p \in N - {1},

the coefficient of

x^{3}

\frac{(1 - x)^{- 1 / p}}{(1 - x)^{n}} =

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

When

| x | < \frac{1}{2},

the coefficient of

x^{4}

in the expansion of

\frac{3 x^{2} - 5 x + 3}{(x - 1) (2 x + 1) (x + 3)}

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

The

13^{th}

term in the expansion of

(1 - 4 x)^{- 4}

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

The coefficient of

x^{n},

where

n

is any positive integer, in the expansion of

{(1 + 2 x + 3 x^{2} + \dots \infty)}^{1 / 2}

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

If the third term in the binomial expansion of

{(1 + x^{\log_{2} x})}^{5}

equals

2560,

then a possible value of

x

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Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

The first negative coefficient in the terms occurring in the expansion of

(1 + x)^{\frac{21}{5}}

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

Let

a > b > 0

and

I (n) = a^{1 / n} - b^{1 / n}, J (n) = (a - b)^{1 / n}

for all

n \geq 2 .

Then

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Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

For

|x| < 1

, the constant term in the expansion of

\frac{1}{{(x - 1)}^{2} (x - 2)}

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

If the coefficient of

x^{13}

in the expansion of

\frac{{(1 + x)}^{2}}{{(1 - 2 x)}^{3}}

A \times 2^{10}

, then

A =

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Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

f (x) = x^{n}

, then the value of

f (1) - \frac{f^{'} (1)}{1!} + \frac{f^{''} (1)}{2!} - \frac{f^{'''} (1)}{3!} + \dots + \frac{{(- 1)}^{n} f^{n} (1)}{n!}

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

For

x > 0,

p^{th}

term is the first negative term in the expansion of

{(1 + \frac{3 x}{5})}^{22 / 3}

and in the expansion of

{(1 - \frac{3 x}{5})}^{22 / 3}

from

r^{t h}

term onwards all the terms are positive, then the number of terms in the expansion of

{(p x + \frac{r}{x})}^{p r}

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

The sum of coefficients of integral powers of

x

in the binomial expansion of

{(1 - 2 \sqrt{x})}^{50}

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Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

\sum_{r = 0}^{25} \{({}^{50}C_{r}) ({}^{50 - r}C_{25 - r})\} = K ({}^{50}C_{25})

, then

K

is equal to

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Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

The positive value of

λ

for which the co-efficient of

x^{2}

in the expansion

x^{2} {(\sqrt{x} + \frac{λ}{x^{2}})}^{10}

720,

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Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

1 + \frac{1}{3} + \frac{1 \cdot 3}{3 \cdot 6} + \frac{1 \cdot 3 \cdot 5}{3 \cdot 6 \cdot 9} + \dots \dots \infty =

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

The number of rational terms in the expansion of

{(3^{\frac{1}{4}} + 7^{\frac{1}{6}})}^{144}

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Any Index

y = \frac{3}{4} + \frac{3 \cdot 5}{4 \cdot 8} + \frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12} + \dots \infty

then

The first negative term in the expansion of 1+x27/5 (0<x<1) is

Important Questions on Binomial Theorem

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The first negative term in the expansion of ${(1 + x)}^{27 / 5}$ $(0 < x < 1)$ is