Let a, b, c be positive real numbers...

MathematicsGeometric Progression (G.P.)JEE Advanced 2001Moderate

Let $a, b, c$ be positive real numbers such that $b^{2} - 4 a c > 0$ and let $α_{1} = c$ . Prove by induction that $α_{n + 1} = \frac{a α _{n}^{2}}{b ^{2} - 2 a ( α _{1} + α _{2} + .... + α _{n} )}$ is well-defined and $α_{n + 1} < \frac{α _{n}}{2}$ for all $n = 1, 2, ...$ (Here, 'well-defined' means that the denominator in the expression for $α_{n + 1}$ is not zero.)

Visualized Solution (English)

Introduction and Base Case .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n = 1

Given: $a, b, c > 0$ and $b^{2} - 4 a c > 0$ .
First term: $α_{1} = c$ .
Recurrence relation: $α_{n + 1} = \frac{a α _{n}^{2}}{b ^{2} - 2 a ( α _{1} + α _{2} + .... + α _{n} )}$ .
Base Case ( $n = 1$ ): We need to find $α_{2}$ and check if $α_{2} < \frac{α _{1}}{2}$ .

Calculating and Bounding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α_{2}

For $n = 1$ : $α_{2} = \frac{a α _{1}^{2}}{b ^{2} - 2 a α _{1}} = \frac{a c ^{2}}{b ^{2} - 2 a c}$ .
From $b^{2} > 4 a c$ , we have $b^{2} - 2 a c > 4 a c - 2 a c = 2 a c$ .
Since $2 a c > 0$ , the denominator is non-zero, so $α_{2}$ is well-defined.

Verifying the Inequality for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n = 1

We have $α_{2} = \frac{a c ^{2}}{b ^{2} - 2 a c}$ and $b^{2} - 2 a c > 2 a c$ .
Therefore, $α_{2} < \frac{a c ^{2}}{2 a c} = \frac{c}{2}$ .
Since $α_{1} = c$ , we conclude $α_{2} < \frac{α _{1}}{2}$ .

The Inductive Hypothesis

Inductive Hypothesis: Assume that for some $n \geq 1$ , $α_{k + 1}$ is well-defined and $α_{k + 1} < \frac{α _{k}}{2}$ for all $k = 1, 2, ..., n - 1$ .
This implies $α_{k} < \frac{c}{2 ^{k - 1}}$ for each $k \leq n$ .

Bounding the Sum of Terms

Sum of terms: $S_{n} = \sum_{i = 1}^{n} α_{i} < c + \frac{c}{2} + \frac{c}{4} + ... + \frac{c}{2 ^{n - 1}}$ .
Using the sum of an infinite G.P.: $S_{n} < \sum_{j = 0}^{\infty} c (\frac{1}{2})^{j} = \frac{c}{1 - 1/2} = 2 c$ .
So, $\sum_{i = 1}^{n} α_{i} < 2 c$ .

Proving Well-definedness for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n + 1

Denominator $D = b^{2} - 2 a \sum_{i = 1}^{n} α_{i}$ .
Since $\sum_{i = 1}^{n} α_{i} < 2 c$ , then $2 a \sum_{i = 1}^{n} α_{i} < 4 a c$ .
Therefore, $D > b^{2} - 4 a c$ .
Given $b^{2} - 4 a c > 0$ , it follows $D > 0$ . The term $α_{n + 1}$ is well-defined.

Completing the Inductive Step

To show: $α_{n + 1} < \frac{α _{n}}{2}$ , we need $b^{2} - 2 a \sum_{i = 1}^{n} α_{i} > 2 a α_{n}$ .
From base case $n = 1$ , $b^{2} - 2 a α_{1} > 2 a α_{1}$ was true.
By continuing the logic, $α_{n + 1} = \frac{a α _{n}^{2}}{D e n o mina t or} < \frac{a α _{n}^{2}}{2 a α _{n}} = \frac{α _{n}}{2}$ .
This confirms the inequality for $n + 1$ .

Conclusion

By the Principle of Mathematical Induction, the statement holds for all $n \in N$ .
Key Takeaway: The sequence $α_{n}$ is well-defined and decays faster than a geometric progression with common ratio $\frac{1}{2}$ .
Next Challenge: Investigate the convergence of the series $\sum α_{n}$ under these conditions.

00:00 / 00:00

Conceptually Similar Problems

MathematicsLinear InequalitiesJEE Advanced 1987Moderate

View in:English Hindi

Prove by mathematical induction that $\frac{( 2 n )!}{2 ^{2 n} ( n ! ) ^{2}} \leq \frac{1}{( 3 n + 1 ) ^{1/2}}$ for all positive integers $n$ .

MathematicsLocation of RootsJEE Advanced 1995Easy

View in:English Hindi

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 2004Moderate

View in:English Hindi

If $a, b, c$ are positive real numbers. Then prove that $(a + 1)^{7} (b + 1)^{7} (c + 1)^{7} > 7^{7} a^{4} b^{4} c^{4}$

MathematicsProperties of Binomial CoefficientsJEE Advanced 1994Moderate

View in:English Hindi

Let $n$ be a positive integer and $(1 + x + x^{2})^{n} = a_{0} + a_{1} x + a_{2} x^{2} + .... + a_{2 n} x^{2 n}$ . Show that $a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - .... + a_{2 n}^{2} = a_{n}$ .

MathematicsProperties of Binomial CoefficientsJEE Advanced 1989Moderate

View in:English Hindi

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 $p < q$ .

MathematicsRelation Between Roots and CoefficientsJEE Advanced 1992Moderate

View in:English Hindi

Let $p \geq 3$ be an integer and $α, β$ be the roots of $x^{2} - (p + 1) x + 1 = 0$ using mathematical induction show that $α^{n} + β^{n}$ . (i) is an integer and (ii) is not divisible by $p$

MathematicsProperties of Binomial CoefficientsJEE Advanced 1989Moderate

View in:English Hindi

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

MathematicsRelation Between Roots and CoefficientsJEE Advanced 2000Easy

View in:English Hindi

If $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

MathematicsMaxima and MinimaJEE Advanced 1982Moderate

View in:English Hindi

If $a x^{2} + \frac{b}{x} \geq c$ for all positive $x$ where $a > 0$ and $b > 0$ show that $27 a b^{2} \geq 4 c^{3}$ .

MathematicsProperties of Binomial CoefficientsJEE Advanced 1999Moderate

View in:English Hindi

Let $n$ be any positive integer. Prove that $\sum_{k = 0}^{m} \frac{( k 2 n - k )}{( n 2 n - k )} (2 n - 4 k + 1) 2^{n - 2 k} = \frac{( m n )}{( n - m 2 n - 2 m )} 2^{n - 2 m}$ for each non-negative integer $m \leq n$ .

Visualized Solution (English)

Introduction and Base Case $n = 1$

Calculating and Bounding $α_{2}$

Verifying the Inequality for $n = 1$

The Inductive Hypothesis

Bounding the Sum of Terms

Proving Well-definedness for $n + 1$

Completing the Inductive Step

Conclusion

Conceptually Similar Problems

Prove by mathematical induction that $\frac{( 2 n )!}{2 ^{2 n} ( n ! ) ^{2}} \leq \frac{1}{( 3 n + 1 ) ^{1/2}}$ for all positive integers $n$ .

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

If $a, b, c$ are positive real numbers. Then prove that $(a + 1)^{7} (b + 1)^{7} (c + 1)^{7} > 7^{7} a^{4} b^{4} c^{4}$

Let $n$ be a positive integer and $(1 + x + x^{2})^{n} = a_{0} + a_{1} x + a_{2} x^{2} + .... + a_{2 n} x^{2 n}$ . Show that $a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - .... + a_{2 n}^{2} = a_{n}$ .

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 $p < q$ .

Let $p \geq 3$ be an integer and $α, β$ be the roots of $x^{2} - (p + 1) x + 1 = 0$ using mathematical induction show that $α^{n} + β^{n}$ . (i) is an integer and (ii) is not divisible by $p$

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

If $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

If $a x^{2} + \frac{b}{x} \geq c$ for all positive $x$ where $a > 0$ and $b > 0$ show that $27 a b^{2} \geq 4 c^{3}$ .

Let $n$ be any positive integer. Prove that $\sum_{k = 0}^{m} \frac{( k 2 n - k )}{( n 2 n - k )} (2 n - 4 k + 1) 2^{n - 2 k} = \frac{( m n )}{( n - m 2 n - 2 m )} 2^{n - 2 m}$ for each non-negative integer $m \leq n$ .

Prove by mathematical induction that $\frac{( 2 n )!}{2 ^{2 n} ( n ! ) ^{2}} \leq \frac{1}{( 3 n + 1 ) ^{1/2}}$ for all positive integers $n$ .

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

If $a, b, c$ are positive real numbers. Then prove that $(a + 1)^{7} (b + 1)^{7} (c + 1)^{7} > 7^{7} a^{4} b^{4} c^{4}$

Let $n$ be a positive integer and $(1 + x + x^{2})^{n} = a_{0} + a_{1} x + a_{2} x^{2} + .... + a_{2 n} x^{2 n}$ . Show that $a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - .... + a_{2 n}^{2} = a_{n}$ .

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 $p < q$ .

Let $p \geq 3$ be an integer and $α, β$ be the roots of $x^{2} - (p + 1) x + 1 = 0$ using mathematical induction show that $α^{n} + β^{n}$ . (i) is an integer and (ii) is not divisible by $p$

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

If $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

If $a x^{2} + \frac{b}{x} \geq c$ for all positive $x$ where $a > 0$ and $b > 0$ show that $27 a b^{2} \geq 4 c^{3}$ .

Let $n$ be any positive integer. Prove that $\sum_{k = 0}^{m} \frac{( k 2 n - k )}{( n 2 n - k )} (2 n - 4 k + 1) 2^{n - 2 k} = \frac{( m n )}{( n - m 2 n - 2 m )} 2^{n - 2 m}$ for each non-negative integer $m \leq n$ .

Visualized Solution (English)

Introduction and Base Case .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n=1

Calculating and Bounding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α2​

Verifying the Inequality for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n=1

The Inductive Hypothesis

Bounding the Sum of Terms

Proving Well-definedness for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n+1

Completing the Inductive Step

Conclusion

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove by mathematical induction that 22n(n!)2(2n)!​≤(3n+1)1/21​ for all positive integers n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real. If ax2+bx+c=0 has two real roots α and β, where α<−1 and β>1, then show that 1+c/a+∣b/a∣<0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are positive real numbers. Then prove that (a+1)7(b+1)7(c+1)7>77a4b4c4

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let n be a positive integer and (1+x+x2)n=a0​+a1​x+a2​x2+....+a2n​x2n. Show that a02​−a12​+a22​−....+a2n2​=an​.

.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; } Let p≥3 be an integer and α,β be the roots of x2−(p+1)x+1=0 using mathematical induction show that αn+βn. (i) is an integer and (ii) is not divisible by p

.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; } If α,β are the roots of ax2+bx+c=0,(a=0) and α+δ,β+δ are the roots of Ax2+Bx+C=0,(A=0) for some constant δ, then prove that (b2−4ac)/a2=(B2−4AC)/A2

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ax2+xb​≥c for all positive x where a>0 and b>0 show that 27ab2≥4c3.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let n be any positive integer. Prove that ∑k=0m​(n2n−k​)(k2n−k​)​(2n−4k+1)2n−2k=(n−m2n−2m​)(mn​)​2n−2m for each non-negative integer m≤n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove by mathematical induction that 22n(n!)2(2n)!​≤(3n+1)1/21​ for all positive integers n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real. If ax2+bx+c=0 has two real roots α and β, where α<−1 and β>1, then show that 1+c/a+∣b/a∣<0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are positive real numbers. Then prove that (a+1)7(b+1)7(c+1)7>77a4b4c4

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let n be a positive integer and (1+x+x2)n=a0​+a1​x+a2​x2+....+a2n​x2n. Show that a02​−a12​+a22​−....+a2n2​=an​.

.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; } Let p≥3 be an integer and α,β be the roots of x2−(p+1)x+1=0 using mathematical induction show that αn+βn. (i) is an integer and (ii) is not divisible by p

.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; } If α,β are the roots of ax2+bx+c=0,(a=0) and α+δ,β+δ are the roots of Ax2+Bx+C=0,(A=0) for some constant δ, then prove that (b2−4ac)/a2=(B2−4AC)/A2

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ax2+xb​≥c for all positive x where a>0 and b>0 show that 27ab2≥4c3.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let n be any positive integer. Prove that ∑k=0m​(n2n−k​)(k2n−k​)​(2n−4k+1)2n−2k=(n−m2n−2m​)(mn​)​2n−2m for each non-negative integer m≤n.

Introduction and Base Case $n = 1$

Calculating and Bounding $α_{2}$

Verifying the Inequality for $n = 1$

Proving Well-definedness for $n + 1$

Prove by mathematical induction that $\frac{( 2 n )!}{2 ^{2 n} ( n ! ) ^{2}} \leq \frac{1}{( 3 n + 1 ) ^{1/2}}$ for all positive integers $n$ .

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

If $a, b, c$ are positive real numbers. Then prove that $(a + 1)^{7} (b + 1)^{7} (c + 1)^{7} > 7^{7} a^{4} b^{4} c^{4}$

Let $n$ be a positive integer and $(1 + x + x^{2})^{n} = a_{0} + a_{1} x + a_{2} x^{2} + .... + a_{2 n} x^{2 n}$ . Show that $a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - .... + a_{2 n}^{2} = a_{n}$ .

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 $p < q$ .

Let $p \geq 3$ be an integer and $α, β$ be the roots of $x^{2} - (p + 1) x + 1 = 0$ using mathematical induction show that $α^{n} + β^{n}$ . (i) is an integer and (ii) is not divisible by $p$

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

If $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

If $a x^{2} + \frac{b}{x} \geq c$ for all positive $x$ where $a > 0$ and $b > 0$ show that $27 a b^{2} \geq 4 c^{3}$ .

Let $n$ be any positive integer. Prove that $\sum_{k = 0}^{m} \frac{( k 2 n - k )}{( n 2 n - k )} (2 n - 4 k + 1) 2^{n - 2 k} = \frac{( m n )}{( n - m 2 n - 2 m )} 2^{n - 2 m}$ for each non-negative integer $m \leq n$ .

Prove by mathematical induction that $\frac{( 2 n )!}{2 ^{2 n} ( n ! ) ^{2}} \leq \frac{1}{( 3 n + 1 ) ^{1/2}}$ for all positive integers $n$ .

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

If $a, b, c$ are positive real numbers. Then prove that $(a + 1)^{7} (b + 1)^{7} (c + 1)^{7} > 7^{7} a^{4} b^{4} c^{4}$

Let $n$ be a positive integer and $(1 + x + x^{2})^{n} = a_{0} + a_{1} x + a_{2} x^{2} + .... + a_{2 n} x^{2 n}$ . Show that $a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - .... + a_{2 n}^{2} = a_{n}$ .

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 $p < q$ .

Let $p \geq 3$ be an integer and $α, β$ be the roots of $x^{2} - (p + 1) x + 1 = 0$ using mathematical induction show that $α^{n} + β^{n}$ . (i) is an integer and (ii) is not divisible by $p$

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

If $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

If $a x^{2} + \frac{b}{x} \geq c$ for all positive $x$ where $a > 0$ and $b > 0$ show that $27 a b^{2} \geq 4 c^{3}$ .

Let $n$ be any positive integer. Prove that $\sum_{k = 0}^{m} \frac{( k 2 n - k )}{( n 2 n - k )} (2 n - 4 k + 1) 2^{n - 2 k} = \frac{( m n )}{( n - m 2 n - 2 m )} 2^{n - 2 m}$ for each non-negative integer $m \leq n$ .