Binomial Theorem

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What is the relation between the sum of coefficients of even terms and the sum of coefficients of odd terms in the expansion of (1+x)^n?

Sum of coefficients of even terms is equal to sum of coefficients of odd terms (correct)

Sum of coefficients of even terms is less than sum of coefficients of odd terms

Sum of coefficients of even terms is greater than sum of coefficients of odd terms

Sum of coefficients of even terms is equal to 2^n

What is the value of C_r in terms of C_(r-1) in the expansion of (1+x)^n?

C_r = nC_(r-1)

C_r = rC_(r-1)

C_r = (n-r+1)C_(r-1)/r (correct)

C_r = (n-r+1)C_(r-1)/(r-1)

If (1+x)^n = C_0 + C_1x + C_2x^2 +... + C_nx^n, then what is the value of C_0 + C_1 + C_2 +... + C_n?

2^n (correct)

n!

n^n

n

What is the sum of the product of coefficients in the expansion of (1+x)^n?

2^n Signup and view all the answers

What is the coefficient of x^r in the expansion of (1+x)^n?

n!/(r!(n-r)!) Signup and view all the answers

What is the total number of terms in the expansion of (x + y)^n?

n+1 Signup and view all the answers

If n Cr = n C(n - r), what can be concluded about the coefficients of consecutive terms in the expansion of (x + y)^n?

They are equidistant from the beginning and end Signup and view all the answers

What is the sum of the powers of x and y in each term of the expansion of (x + y)^n?

n Signup and view all the answers

What is the value of n Cr when r = 0?

1 Signup and view all the answers

What is the property of the binomial coefficients n Cr in the expansion of (x + y)^n?

They are equidistant from the beginning and end Signup and view all the answers

If $a_1, a_2, a_3, a_4$ are the coefficients of any four consecutive terms in the expansion of $(1+x)^n$, then what is the value of $\frac{a_1}{a_3}+\frac{a_2}{a_4}$?

$\frac{n+1}{n-1}$ Signup and view all the answers

In the expansion of $(1+x)^n$, if $a_r$ is the coefficient of $x^{r-1}$, then what is the value of $\frac{a_r}{a_{r+2}}$?

$\frac{r+1}{n-r+1}$ Signup and view all the answers

In Pascal's Triangle, each entry is the sum of the two entries directly above it. What is the value of the sum of the reciprocals of all the binomial coefficients in the $(n+1)^{th}$ row?

$2$ Signup and view all the answers

If $a_1, a_2, a_3, a_4$ are the coefficients of any four consecutive terms in the expansion of $(1+x)^n$, then what is the value of $a_1a_3-a_2a_4$?

$0$ Signup and view all the answers

In the expansion of $(1+x)^n$, what is the value of the coefficient of $x^{n-2}$ in terms of $n$?

$\frac{n(n-1)}{2}$ Signup and view all the answers

What is the general term in the expansion of (x + y)^n?

n C_r x^(n - r) y^r Signup and view all the answers

What is the independent term in the expansion of (x + y)^n?

n C_n y^0 Signup and view all the answers

What is the coefficient of the constant term in the expansion of (x + y)^n?

n C_n Signup and view all the answers

What is the sum of the coefficients of the odd terms in the expansion of (x + y)^n?

2^n Signup and view all the answers

What is the relation between the coefficients of the binomial expansion of (x + y)^n and (x - y)^n?

They are equal in magnitude but alternate in sign Signup and view all the answers

What is the value of the middle term in the expansion of (1+x)^2n?

10 C5 x^5 Signup and view all the answers

If the middle term in the expansion of (1+x)^n is 10 C5 x^5, what is the value of n?

10 Signup and view all the answers

In the expansion of (x+y)^n, what is the value of the coefficient of the middle term?

n!/((n/2)!*(n/2)!) Signup and view all the answers

What is the coefficient of x^r in the expansion of (1+x)^n?

n Cr Signup and view all the answers

If the middle term in the expansion of (1+x)^n is T((n+1)/2), what is the value of T((n+1)/2) when n is odd?

n C((n+1)/2) Signup and view all the answers

What is the condition for the validity of the expansion (1 + x)^n?

-1 < x < 1 Signup and view all the answers

Binomial Theorem

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Questions and Answers

What is the relation between the sum of coefficients of even terms and the sum of coefficients of odd terms in the expansion of (1+x)^n?

What is the value of C_r in terms of C_(r-1) in the expansion of (1+x)^n?

If (1+x)^n = C_0 + C_1x + C_2x^2 +... + C_nx^n, then what is the value of C_0 + C_1 + C_2 +... + C_n?

What is the sum of the product of coefficients in the expansion of (1+x)^n?

What is the coefficient of x^r in the expansion of (1+x)^n?

What is the total number of terms in the expansion of (x + y)^n?

If n Cr = n C(n - r), what can be concluded about the coefficients of consecutive terms in the expansion of (x + y)^n?

What is the sum of the powers of x and y in each term of the expansion of (x + y)^n?

What is the value of n Cr when r = 0?

What is the property of the binomial coefficients n Cr in the expansion of (x + y)^n?

If $a_1, a_2, a_3, a_4$ are the coefficients of any four consecutive terms in the expansion of $(1+x)^n$, then what is the value of $\frac{a_1}{a_3}+\frac{a_2}{a_4}$?

In the expansion of $(1+x)^n$, if $a_r$ is the coefficient of $x^{r-1}$, then what is the value of $\frac{a_r}{a_{r+2}}$?

In Pascal's Triangle, each entry is the sum of the two entries directly above it. What is the value of the sum of the reciprocals of all the binomial coefficients in the $(n+1)^{th}$ row?

If $a_1, a_2, a_3, a_4$ are the coefficients of any four consecutive terms in the expansion of $(1+x)^n$, then what is the value of $a_1a_3-a_2a_4$?

In the expansion of $(1+x)^n$, what is the value of the coefficient of $x^{n-2}$ in terms of $n$?

What is the general term in the expansion of (x + y)^n?

What is the independent term in the expansion of (x + y)^n?

What is the coefficient of the constant term in the expansion of (x + y)^n?

What is the sum of the coefficients of the odd terms in the expansion of (x + y)^n?

What is the relation between the coefficients of the binomial expansion of (x + y)^n and (x - y)^n?

What is the value of the middle term in the expansion of (1+x)^2n?

If the middle term in the expansion of (1+x)^n is 10 C5 x^5, what is the value of n?

In the expansion of (x+y)^n, what is the value of the coefficient of the middle term?

What is the coefficient of x^r in the expansion of (1+x)^n?

If the middle term in the expansion of (1+x)^n is T((n+1)/2), what is the value of T((n+1)/2) when n is odd?

What is the condition for the validity of the expansion (1 + x)^n?

What is the value of the coefficient of a^3b^4c^5 in the expansion of (bc + ca + ab)^6?

What is the value of n in the expansion (1 + x)^n when x is very small?

What is the number of terms in the series expansion of (1 + x)^n?

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