Binomial Theorem Questions

MathematicsBinomial Theorem69 Questions Found

MathematicsProperties of Binomial CoefficientsJEE Advanced 1978Moderate

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If $f (m, n) = \frac{( 1 - x ^{m} ) ( 1 - x ^{m - 1} ) \dots ( 1 - x ^{m - n + 1} )}{( 1 - x ) ( 1 - x ^{2} ) \dots ( 1 - x ^{n} )}$ where $m$ and $n$ are positive integers ( $n \leq m$ ), show that $f (m, n + 1) = f (m - 1, n + 1) + x^{m - n - 1} f (m - 1, n)$ .

MathematicsProperties of Binomial CoefficientsJEE Advanced 1979Moderate

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Given that $C_{1} + 2 C_{2} x + 3 C_{3} x^{2} + .... + 2 n C_{2 n} x^{2 n - 1} = 2 n (1 + x)^{2 n - 1}$ where $C_{r} = \frac{( 2 n )!}{r ! ( 2 n - r )!}, r = 0, 1, 2, ...., 2 n$ . Prove that $C_{1}^{2} - 2 C_{2}^{2} + 3 C_{3}^{2} - .... - 2 n C_{2 n}^{2} = (- 1)^{n} n^{n} C_{n}$ .

MathematicsBinomial Expansion for Positive Integral IndexJEE Advanced 1982Easy

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Prove that $7^{2 n} + 2^{3 n - 3} (3^{n - 1})$ is divisible by 25 for any natural number $n$ .

MathematicsProperties of Binomial CoefficientsJEE Advanced 1983Easy

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If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + .... + C_{n} x^{n}$ then show that the sum of the products of the $C_{i}$ 's taken two at a time, represented by $\sum_{0 \leq i < j \leq n} C_{i} C_{j}$ is equal to $2^{2 n - 1} - \frac{( 2 n )!}{2 ( n ! ) ^{2}}$

MathematicsProperties of Binomial CoefficientsJEE Advanced 1984Moderate

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Given $s_{n} = 1 + q + q^{2} + .... + q^{n}$ ; $S_{n} = 1 + \frac{q + 1}{2} + (\frac{q + 1}{2})^{2} + .... + (\frac{q + 1}{2})^{n}, q \neq = 1$ Prove that $^{n + 1} C_{1} +^{n + 1} C_{2} s_{1} +^{n + 1} C_{3} s_{2} + .... +^{n + 1} C_{n + 1} s_{n} = 2^{n} S_{n}$

MathematicsBinomial Expansion for Positive Integral IndexJEE Advanced 1984Easy

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If $p$ be a natural number then prove that $p^{n + 1} + (p + 1)^{2 n - 1}$ is divisible by $p^{2} + p + 1$ for every positive integer $n$ .

MathematicsBinomial Expansion for Positive Integral IndexJEE Advanced 1988Moderate

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Let $R = (55 + 11)^{2 n + 1}$ and $f = R - [R]$ , where $[]$ denotes the greatest integer function. Prove that $R f = 4^{2 n + 1}$ .

MathematicsProperties of Binomial CoefficientsJEE Advanced 1989Moderate

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Prove that $C_{0} - 2^{2} C_{1} + 3^{2} C_{2} - .... + (- 1)^{n} (n + 1)^{2} C_{n} = 0, n > 2$ , where $C_{r} =^{n} C_{r}$ .

MathematicsProperties of Binomial CoefficientsJEE Advanced 1989Moderate

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Using mathematical induction, prove that $^{m} C_{0}^{n} C_{k} +^{m} C_{1}^{n} C_{k - 1} + .... +^{m} C_{k}^{n} C_{0} =^{(m + n)} C_{k}$ , where $m, n, k$ are positive integers, and $^{p} C_{q} = 0$ for $p < q$ .

MathematicsBinomial Expansion for Positive Integral IndexJEE Advanced 1990Moderate

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Prove that $\frac{n ^{7}}{7} + \frac{n ^{5}}{5} + \frac{2 n ^{3}}{3} - \frac{n}{105}$ is an integer for every positive integer $n$ .

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that 72n+23n−3(3n−1) is divisible by 25 for any natural number n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given sn​=1+q+q2+....+qn; Sn​=1+2q+1​+(2q+1​)2+....+(2q+1​)n,q=1 Prove that n+1C1​+n+1C2​s1​+n+1C3​s2​+....+n+1Cn+1​sn​=2nSn​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If p be a natural number then prove that pn+1+(p+1)2n−1 is divisible by p2+p+1 for every positive integer n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R=(55​+11)2n+1 and f=R−[R], where [ ] denotes the greatest integer function. Prove that Rf=42n+1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that C0​−22C1​+32C2​−....+(−1)n(n+1)2Cn​=0,n>2, where Cr​=nCr​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Using mathematical induction, prove that mC0n​Ck​+mC1n​Ck−1​+....+mCkn​C0​=(m+n)Ck​, where m,n,k are positive integers, and pCq​=0 for p<q.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that 7n7​+5n5​+32n3​−105n​ is an integer for every positive integer n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that 72n+23n−3(3n−1) is divisible by 25 for any natural number n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given sn​=1+q+q2+....+qn; Sn​=1+2q+1​+(2q+1​)2+....+(2q+1​)n,q=1 Prove that n+1C1​+n+1C2​s1​+n+1C3​s2​+....+n+1Cn+1​sn​=2nSn​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If p be a natural number then prove that pn+1+(p+1)2n−1 is divisible by p2+p+1 for every positive integer n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R=(55​+11)2n+1 and f=R−[R], where [ ] denotes the greatest integer function. Prove that Rf=42n+1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that C0​−22C1​+32C2​−....+(−1)n(n+1)2Cn​=0,n>2, where Cr​=nCr​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Using mathematical induction, prove that mC0n​Ck​+mC1n​Ck−1​+....+mCkn​C0​=(m+n)Ck​, where m,n,k are positive integers, and pCq​=0 for p<q.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that 7n7​+5n5​+32n3​−105n​ is an integer for every positive integer n.

Prove that $7^{2 n} + 2^{3 n - 3} (3^{n - 1})$ is divisible by 25 for any natural number $n$ .

Given $s_{n} = 1 + q + q^{2} + .... + q^{n}$ ; $S_{n} = 1 + \frac{q + 1}{2} + (\frac{q + 1}{2})^{2} + .... + (\frac{q + 1}{2})^{n}, q \neq = 1$ Prove that $^{n + 1} C_{1} +^{n + 1} C_{2} s_{1} +^{n + 1} C_{3} s_{2} + .... +^{n + 1} C_{n + 1} s_{n} = 2^{n} S_{n}$

If $p$ be a natural number then prove that $p^{n + 1} + (p + 1)^{2 n - 1}$ is divisible by $p^{2} + p + 1$ for every positive integer $n$ .

Let $R = (55 + 11)^{2 n + 1}$ and $f = R - [R]$ , where $[]$ denotes the greatest integer function. Prove that $R f = 4^{2 n + 1}$ .

Prove that $C_{0} - 2^{2} C_{1} + 3^{2} C_{2} - .... + (- 1)^{n} (n + 1)^{2} C_{n} = 0, n > 2$ , where $C_{r} =^{n} C_{r}$ .

Using mathematical induction, prove that $^{m} C_{0}^{n} C_{k} +^{m} C_{1}^{n} C_{k - 1} + .... +^{m} C_{k}^{n} C_{0} =^{(m + n)} C_{k}$ , where $m, n, k$ are positive integers, and $^{p} C_{q} = 0$ for $p < q$ .

Prove that $\frac{n ^{7}}{7} + \frac{n ^{5}}{5} + \frac{2 n ^{3}}{3} - \frac{n}{105}$ is an integer for every positive integer $n$ .

Prove that $7^{2 n} + 2^{3 n - 3} (3^{n - 1})$ is divisible by 25 for any natural number $n$ .

Given $s_{n} = 1 + q + q^{2} + .... + q^{n}$ ; $S_{n} = 1 + \frac{q + 1}{2} + (\frac{q + 1}{2})^{2} + .... + (\frac{q + 1}{2})^{n}, q \neq = 1$ Prove that $^{n + 1} C_{1} +^{n + 1} C_{2} s_{1} +^{n + 1} C_{3} s_{2} + .... +^{n + 1} C_{n + 1} s_{n} = 2^{n} S_{n}$

If $p$ be a natural number then prove that $p^{n + 1} + (p + 1)^{2 n - 1}$ is divisible by $p^{2} + p + 1$ for every positive integer $n$ .

Let $R = (55 + 11)^{2 n + 1}$ and $f = R - [R]$ , where $[]$ denotes the greatest integer function. Prove that $R f = 4^{2 n + 1}$ .

Prove that $C_{0} - 2^{2} C_{1} + 3^{2} C_{2} - .... + (- 1)^{n} (n + 1)^{2} C_{n} = 0, n > 2$ , where $C_{r} =^{n} C_{r}$ .

Using mathematical induction, prove that $^{m} C_{0}^{n} C_{k} +^{m} C_{1}^{n} C_{k - 1} + .... +^{m} C_{k}^{n} C_{0} =^{(m + n)} C_{k}$ , where $m, n, k$ are positive integers, and $^{p} C_{q} = 0$ for $p < q$ .

Prove that $\frac{n ^{7}}{7} + \frac{n ^{5}}{5} + \frac{2 n ^{3}}{3} - \frac{n}{105}$ is an integer for every positive integer $n$ .