Binomial Theorem Questions (605)

If the middle term of \(\left(\frac{1}{x} + x \sin x\right)^{10}\) is equal to \(7\frac{7}{8}\), then value of \(x\) is

If in the expansion of \((1+x)^n\), \(a, b, c\) are three consecutive coefficients, then \(n =\)

If \(x > 0\), the first negative term in the expansion of \((1+x)^{27/5}\) is:

The sum of the last eight coefficients in the expansion of \((1 + x)^{16}\) is equal to

The value of \(\dfrac{{}^nC_0}{n} + \dfrac{{}^nC_1}{n+1} + \dfrac{{}^nC_2}{n+2} + \cdots + \dfrac{{}^nC_n}{2n}\) is equal to

If n is odd, the value of ∑r=1(n+1)/2 a2r-1 is

The sum of the coefficients of all odd degree terms in the expansion of \(\left(x + \sqrt{x^3 - 1}\right)^5 + \left(x - \sqrt{x^3 - 1}\right)^5\), \((x > 1)\) is

For \(r = 0, 1, \ldots, 10\), let \(A_r\), \(B_r\), and \(C_r\) denote, respectively, the coefficients of \(x^r\) in the expansions of \((1+x)^{10}\), \((1+x)^{20}\) and \((1+x)^{30}\). Then \(\displaystyle\sum_{r=1}^{10} A_r(B_{10}B_r - C_{10}A_r)\) is equal to

In the expansion of \(\left(x^3 - \dfrac{1}{x^2}\right)^{15}\), the constant term is

The term independent of \(x\) in \((1+x)^m \left(1 + \frac{1}{x}\right)^n\) is

The coefficient of \(x^9\) in the expansion of \((1+x)(1+x^2)(1+x^3)\cdots(1+x^{100})\) is _______.

If \((1 - x + x^2)^n = a_0 + a_1 x + a_2 x^2 + \cdots + a_{2n} x^{2n}\), then \(a_0 + a_2 + a_4 + \cdots + a_{2n}\) is equal to

Find the value of \((\sqrt{2} + 1)^6 - (\sqrt{2} - 1)^6\).

Find the value of \(\dfrac{18^3 + 7^3 + 3 \times 18 \times 7 \times 25}{3^6 + 6/243 \times 2 + 15/81 \times 4 + 20/27 \times 8 + 15/9 \times 16 + 6/3 \times 32 + 64}\).

For Problems 7–9: An equation \(a_0 + a_1 x + a_2 x^2 + \cdots + a_{99} x^{99} + x^{100} = 0\) has roots \({}^{99}C_0, {}^{99}C_1, {}^{99}C_2, \ldots, {}^{99}C_{99}\).9. The value of \(({}^{99}C_0)^2 + ({}^{99}C_1)^2 + \cdots + ({}^{99}C_{99})^2\) is equal to

If \((1-x^2)^n = \displaystyle\sum_{r=0}^{n} a_r x^r (1-x)^{2n-r}\), then \(a_r\) is equal to

The sum of the real values of \(x\) for which the middle term in the binomial expansion of \(\left(\frac{x^3}{3} + \frac{3}{x}\right)^8\) equals 5670 is

If \(C_r = {}^nC_r\), the sum of the series \(\dfrac{2\left[\left(\dfrac{n}{2}\right)!\right]^2}{n!}\left[C_0^2 - 2C_1^2 + 3C_2^2 - \ldots + (-1)^n(n+1)C_n^2\right]\), where \(n\) is an even integer, is

In the binomial expansion of \((a - b)^n\), \(n \geq 5\), the sum of 5th and 6th terms is zero, then \(\dfrac{a}{b}\) equals

The remainder when the number \(3^{256} - 3^{12}\) is divided by 8 is

If \(\left(2+\dfrac{x}{3}\right)^{55}\) is expanded in the ascending powers of \(x\) and the coefficients of powers of \(x\) in two consecutive terms of the expansion are equal, then these terms are

The value of (21C1 × 10C1) + (21C2 × 10C2) + (21C3 × 10C3) + (21C4 × 10C4) + ... + (21C10 × 10C10) is

The term independent of x in the expansion of \[\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)^{10}\] is:

In the expansion of \((1 + 3x + 2x^2)^6\), find the coefficient of x11.

If \((1-x)^{-n} = a_0 + a_1 x + a_2 x^2 + \cdots + a_r x^r + \cdots\), then \(a_0 + a_1 + a_2 + \cdots + a_r\) is equal to

The coefficient of \(x^{1012}\) in the expansion of \((1+x^n+x^{253})^{10}\), (where \(n\leq 22\) is any positive integer), is

Find the remainder when \(7^{103}\) is divided by 25.

The total number of irrational terms in the binomial expansion of \((7^{1/5} - 3^{1/10})^{60}\) is:

A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of \(\left(2^{1/3} + \dfrac{1}{2(3)^{1/3}}\right)^{10}\) is:

If Tr+1 is the general term of \(\left[ax^2 + \frac{1}{bx}\right]^{11}\), and the coefficient of \(x^7\) in \(\left[ax^2 + \frac{1}{bx}\right]^{11}\) equals the coefficient of \(x^{-7}\) in \(\left[ax - \frac{1}{bx^2}\right]^{11}\), then \(ab\) equals:

The coefficient of the middle term in the binomial expansion in powers of \(x\) of \((1 + \alpha x)^4\) and of \((1 - \alpha x)^6\) is the same, if \(\alpha\) equals

\(1 + \dfrac{1}{4} + \dfrac{1 \times 3}{4 \times 8} + \dfrac{1 \times 3 \times 5}{4 \times 8 \times 12} + \cdots =\)

If \(\displaystyle\sum_{r=0}^{n}\{a_r(x-\alpha+2)^r - b_r(\alpha-x-1)^r\} = 0\), then \(b_n\) is

Consider (1 + x + x2)n = ∑arxr, where a0, a1, a2, ..., a2n are real numbers and n is a positive integer. The value of ∑r=0n rar is

If \(x\) is so small that \(x^3\) and higher powers of \(x\) may be neglected, then \(\dfrac{(1+x)^{3/2} - \left(1 + \dfrac{1}{2}x\right)^3}{(1-x)^{1/2}}\) may be approximated as

If \(10^m\) divides the number \(101^{100} - 1\), then find the greatest value of \(m\).

If \(\dfrac{1}{\sqrt{4x+1}}\left\{\left(\dfrac{1+\sqrt{4x+1}}{2}\right)^n - \left(\dfrac{1-\sqrt{4x+1}}{2}\right)^n\right\} = a_0 + a_1 x + \cdots + a_5 x^5\), then find the possible values of \(n\).

If \((4x^2 + 1)^n = \displaystyle\sum_{r=0}^{n} a_r(1+x^2)^{n-r} x^{2r}\), then the value of \(\displaystyle\sum_{r=0}^{n} a_r\) is

If the coefficients of \(r^{\text{th}}\), \((r+1)^{\text{th}}\) and \((r+2)^{\text{th}}\) terms in the binomial expansion of \((1+y)^m\) are in AP, then \(m\) and \(r\) satisfy the equation

Find the middle term in the expansion of \(\left(x^2 + \dfrac{1}{x^2} + 2\right)^n\).

The coefficient of \(x^{50}\) in the expansion of \((1+x)^{1000} + 2x(1+x)^{999} + 3x^2(1+x)^{998} + \cdots + 1001 x^{1000}\) is:

For Problems 15–17: Let \(P = \displaystyle\sum_{r=1}^{50} \frac{{}^{50+r}C_r(2r-1)}{{}^{50}C_r(50+r)}\), \(Q = \displaystyle\sum_{r=0}^{50} \left({}^{50}C_r\right)^2\), \(R = \displaystyle\sum_{r=0}^{100} (-1)^r \left({}^{100}C_r\right)^2\)16. The value of \(P - R\) is equal to

Find the coefficient of \(x^7\) in the expansion of \((1 + 3x - 2x^3)^{10}\).

If the sum of binomial coefficients in the expansion of \((x + y)^n\) is 1024 then the greatest binomial coefficient occurs in the ______ th term.

\(\binom{30}{0} - \binom{30}{10} + \binom{30}{1} - \binom{30}{11} + \ldots + \binom{30}{20} - \binom{30}{30}\) is equal to

\(\sum_{r=0}^{n} \frac{(-1)^r(\binom{n}{r})^2}{1+nx}\) is equal to

The coefficient of x-5 in the binomial expansion of \(\left(\frac{x \ + \ 1}{x^{\frac{2}{3}} \ - \ x^{\frac{1}{3}} \ + \ 1}-\frac{x \ - \ 1}{x \ - \ x^{\frac{1}{2}}}\right)^{10}\) where x \(\ne\) 0, 1, is:

If f(x) = \sum_{r=1}^{n} \left\{ r^2 \left( \binom{n}{r} + \binom{n}{r+1} \right) - (2r-1)\binom{n}{r} \right\} and f(30) = 30(2)^5, then the value of n is

If the rth term in the expansion of \((1 + x)^{20}\) has its coefficient equal to that of the \((r + 4)\)th term, then \(r\) is

If in the expansion of \(\left(\frac{1}{\sqrt[3]{2}} + \frac{1}{\sqrt[3]{3}}\right)^{n}\), the ratio of 7th term from the beginning to the 7th term from the end is \(\frac{1}{6}\), then \(n\) is