If a, b, c are in A.P., a², b², c² are...

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 2003Moderate

If $a, b, c$ are in A.P., $a^{2}, b^{2}, c^{2}$ are in H.P., then prove that either $a = b = c$ or $a, b, - c /2$ form a G.P.

Visualized Solution (Hindi)

Defining the A.P. Condition

Given that $a, b, c$ are in A.P.
The common difference is constant: $b - a = c - b$
Rearranging gives the standard A.P. property: $2 b = a + c$

Defining the H.P. Condition

Given that $a^{2}, b^{2}, c^{2}$ are in H.P.
The middle term is the harmonic mean: $b^{2} = \frac{2 a ^{2} c ^{2}}{a ^{2} + c ^{2}}$

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b into the H.P. Equation

From A.P.: $b = \frac{a + c}{2}$
Substitute $b$ into the H.P. equation:
$(\frac{a + c}{2})^{2} = \frac{2 a ^{2} c ^{2}}{a ^{2} + c ^{2}}$

Cross-Multiplication and Simplification

Expand the square: $\frac{( a + c ) ^{2}}{4} = \frac{2 a ^{2} c ^{2}}{a ^{2} + c ^{2}}$
Cross-multiply: $(a + c)^{2} (a^{2} + c^{2}) = 8 a^{2} c^{2}$

Expanding the Binomial

Expand $(a + c)^{2}$ :
$(a^{2} + c^{2} + 2 a c) (a^{2} + c^{2}) = 8 a^{2} c^{2}$

Quadratic Substitution

Let $X = a^{2} + c^{2}$
Substitute $X$ into the equation: $(X + 2 a c) X = 8 a^{2} c^{2}$
Rearrange to quadratic form: $X^{2} + 2 a c X - 8 a^{2} c^{2} = 0$

Factoring the Quadratic

Factor the quadratic: $(X + 4 a c) (X - 2 a c) = 0$
This implies either $X - 2 a c = 0$ or $X + 4 a c = 0$

Case 1: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a = b = c

Case 1: $X - 2 a c = 0$
Substitute $X$ : $a^{2} + c^{2} - 2 a c = 0 \Rightarrow (a - c)^{2} = 0$
This gives $a = c$
Since $2 b = a + c$ , then $2 b = 2 a \Rightarrow a = b = c$

Case 2: Setting up the G.P.

Case 2: $X + 4 a c = 0$
Substitute $X$ : $a^{2} + c^{2} = - 4 a c$
Add $2 a c$ to both sides: $a^{2} + c^{2} + 2 a c = - 4 a c + 2 a c$
Simplify: $(a + c)^{2} = - 2 a c$

Proving the G.P. Condition

Substitute $a + c = 2 b$ into the equation:
$(2 b)^{2} = - 2 a c \Rightarrow 4 b^{2} = - 2 a c$
Divide by $4$ : $b^{2} = a (- \frac{c}{2})$
This is the condition for $a, b, - \frac{c}{2}$ to be in G.P.

Final Conclusion

Conclusion:
Either $a = b = c$
OR $a, b, - \frac{c}{2}$ form a G.P.
The proof is complete.

00:00 / 00:00

Conceptually Similar Problems

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 1988Easy

View in:English Hindi

If the first and the $(2 n - 1)$ st terms of an A.P., a G.P. and an H.P. are equal and their $n$ th terms are $a, b$ and $c$ respectively, then

* Multiple Correct Options

(A)

a = b = c

(B)

a \geq b \geq c

(C)

a + c = b

(D)

a c - b^{2} = 0

Answer:A, B, D

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 1994Easy

View in:English Hindi

If $ln (a + c), ln (a - c), ln (a - 2 b + c)$ are in A.P., then

(A)

a, b, c

are in A.P.

(B)

a^{2}, b^{2}, c^{2}

are in A.P.

(C)

a, b, c

are in G.P.

(D)

a, b, c

are in H.P.

Answer:D

MathematicsHarmonic Progression (H.P.)JEE Main 2005Easy

View in:English Hindi

If $x = \sum_{n = 0}^{\infty} a^{n}, y = \sum_{n = 0}^{\infty} b^{n}, z = \sum_{n = 0}^{\infty} c^{n}$ where $a, b, c$ are in A.P and $∣ a ∣ < 1, ∣ b ∣ < 1, ∣ c ∣ < 1$ then $x, y, z$ are in

(A)

G.P.

(B)

A.P.

(C)

Arithmetic - Geometric Progression

(D)

H.P.

Answer:D

MathematicsCommon RootsJEE Advanced 1985Moderate

View in:English Hindi

If $a, b, c$ are in G.P., then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root if $d / a, e / b, f / c$ are in

(A)

A.P.

(B)

G.P.

(C)

H.P.

(D)

none of these

Answer:A

MathematicsArithmetic Progression (A.P.)JEE Advanced 2002Moderate

View in:English Hindi

Suppose $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. if $a < b < c$ and $a + b + c = 3/2$ , then the value of $a$ is

(A)

\frac{1}{2 2}

(B)

\frac{1}{2 3}

(C)

\frac{1}{2} - \frac{1}{3}

(D)

\frac{1}{2} - \frac{1}{2}

Answer:D

MathematicsProperties of TrianglesJEE Main 2003Easy

View in:English Hindi

If in a $Δ A B C$ $a cos^{2} (\frac{C}{2}) + c cos^{2} (\frac{A}{2}) = \frac{3 b}{2}$ , then the sides $a, b$ and $c$

(A)

satisfy

a + b = c

(B)

are in A.P

(C)

are in G.P

(D)

are in H.P.

Answer:B

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 2002Moderate

View in:English Hindi

Let $a, b$ be positive real numbers. If $a, A_{1}, A_{2}, b$ are in arithmetic progression, $a, G_{1}, G_{2}, b$ are in geometric progression and $a, H_{1}, H_{2}, b$ are in harmonic progression, show that $\frac{G _{1} G _{2}}{H _{1} H _{2}} = \frac{A _{1} + A _{2}}{H _{1} + H _{2}} = \frac{( 2 a + b ) ( a + 2 b )}{9 ab}$ .

MathematicsHarmonic Progression (H.P.)JEE Advanced 2001Easy

View in:English Hindi

Let the positive numbers $a, b, c, d$ be in A.P. Then $ab c, ab d, a c d, b c d$ are

(A)

NOT in A.P./G.P./H.P.

(B)

in A.P.

(C)

in G.P.

(D)

in H.P.

Answer:D

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Main 2003Moderate

View in:English Hindi

If the system of linear equations $x + 2 a y + a z = 0$ ; $x + 3 b y + b z = 0$ ; $x + 4 cy + cz = 0$ ; has a non-zero solution, then $a, b, c$

(A)

satisfy

a + 2 b + 3 c = 0

(B)

are in A.P

(C)

are in G.P

(D)

are in H.P.

Answer:D

MathematicsProperties of TrianglesJEE Advanced 1983Easy

View in:English Hindi

If $a, b, c$ are in A.P., $a^{2}, b^{2}, c^{2}$ are in H.P., then prove that either $a = b = c$ or $a, b, - c /2$ form a G.P.

Visualized Solution (Hindi)

Defining the A.P. Condition

Defining the H.P. Condition

Substituting $b$ into the H.P. Equation

Cross-Multiplication and Simplification

Expanding the Binomial

Quadratic Substitution

Factoring the Quadratic

Case 1: $a = b = c$

Case 2: Setting up the G.P.

Proving the G.P. Condition

Final Conclusion

Conceptually Similar Problems

If the first and the $(2 n - 1)$ st terms of an A.P., a G.P. and an H.P. are equal and their $n$ th terms are $a, b$ and $c$ respectively, then

If $ln (a + c), ln (a - c), ln (a - 2 b + c)$ are in A.P., then

If $x = \sum_{n = 0}^{\infty} a^{n}, y = \sum_{n = 0}^{\infty} b^{n}, z = \sum_{n = 0}^{\infty} c^{n}$ where $a, b, c$ are in A.P and $∣ a ∣ < 1, ∣ b ∣ < 1, ∣ c ∣ < 1$ then $x, y, z$ are in

If $a, b, c$ are in G.P., then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root if $d / a, e / b, f / c$ are in

Suppose $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. if $a < b < c$ and $a + b + c = 3/2$ , then the value of $a$ is

If in a $Δ A B C$ $a cos^{2} (\frac{C}{2}) + c cos^{2} (\frac{A}{2}) = \frac{3 b}{2}$ , then the sides $a, b$ and $c$

Let the positive numbers $a, b, c, d$ be in A.P. Then $ab c, ab d, a c d, b c d$ are

If the system of linear equations $x + 2 a y + a z = 0$ ; $x + 3 b y + b z = 0$ ; $x + 4 cy + cz = 0$ ; has a non-zero solution, then $a, b, c$

The ex-radii $r_{1}, r_{2}, r_{3}$ of $Δ A B C$ are in H.P. Show that its sides $a, b, c$ are in A.P.

If $a, b, c$ are in A.P., $a^{2}, b^{2}, c^{2}$ are in H.P., then prove that either $a = b = c$ or $a, b, - c /2$ form a G.P.

If the first and the $(2 n - 1)$ st terms of an A.P., a G.P. and an H.P. are equal and their $n$ th terms are $a, b$ and $c$ respectively, then

If $ln (a + c), ln (a - c), ln (a - 2 b + c)$ are in A.P., then

If $x = \sum_{n = 0}^{\infty} a^{n}, y = \sum_{n = 0}^{\infty} b^{n}, z = \sum_{n = 0}^{\infty} c^{n}$ where $a, b, c$ are in A.P and $∣ a ∣ < 1, ∣ b ∣ < 1, ∣ c ∣ < 1$ then $x, y, z$ are in

If $a, b, c$ are in G.P., then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root if $d / a, e / b, f / c$ are in

Suppose $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. if $a < b < c$ and $a + b + c = 3/2$ , then the value of $a$ is

If in a $Δ A B C$ $a cos^{2} (\frac{C}{2}) + c cos^{2} (\frac{A}{2}) = \frac{3 b}{2}$ , then the sides $a, b$ and $c$

Let the positive numbers $a, b, c, d$ be in A.P. Then $ab c, ab d, a c d, b c d$ are

If the system of linear equations $x + 2 a y + a z = 0$ ; $x + 3 b y + b z = 0$ ; $x + 4 cy + cz = 0$ ; has a non-zero solution, then $a, b, c$

The ex-radii $r_{1}, r_{2}, r_{3}$ of $Δ A B C$ are in H.P. Show that its sides $a, b, c$ are in A.P.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are in A.P., a2,b2,c2 are in H.P., then prove that either a=b=c or a,b,−c/2 form a G.P.

Visualized Solution (Hindi)

Defining the A.P. Condition

Defining the H.P. Condition

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b into the H.P. Equation

Cross-Multiplication and Simplification

Expanding the Binomial

Quadratic Substitution

Factoring the Quadratic

Case 1: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a=b=c

Case 2: Setting up the G.P.

Proving the G.P. Condition

Final Conclusion

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the first and the (2n−1)st terms of an A.P., a G.P. and an H.P. are equal and their nth terms are a,b and c respectively, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ln(a+c),ln(a−c),ln(a−2b+c) are in A.P., then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x=∑n=0∞​an,y=∑n=0∞​bn,z=∑n=0∞​cn where a,b,c are in A.P and ∣a∣<1,∣b∣<1,∣c∣<1 then x,y,z are in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are in G.P., then the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common root if d/a,e/b,f/c are in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Suppose a,b,c are in A.P. and a2,b2,c2 are in G.P. if a<b<c and a+b+c=3/2, then the value of a is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If in a ΔABC acos2(2C​)+ccos2(2A​)=23b​, then the sides a,b and c

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the positive numbers a,b,c,d be in A.P. Then abc,abd,acd,bcd are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the system of linear equations x+2ay+az=0; x+3by+bz=0; x+4cy+cz=0; has a non-zero solution, then a,b,c

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The ex-radii r1​,r2​,r3​ of ΔABC are in H.P. Show that its sides a,b,c are in A.P.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are in A.P., a2,b2,c2 are in H.P., then prove that either a=b=c or a,b,−c/2 form a G.P.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the first and the (2n−1)st terms of an A.P., a G.P. and an H.P. are equal and their nth terms are a,b and c respectively, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ln(a+c),ln(a−c),ln(a−2b+c) are in A.P., then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x=∑n=0∞​an,y=∑n=0∞​bn,z=∑n=0∞​cn where a,b,c are in A.P and ∣a∣<1,∣b∣<1,∣c∣<1 then x,y,z are in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are in G.P., then the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common root if d/a,e/b,f/c are in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Suppose a,b,c are in A.P. and a2,b2,c2 are in G.P. if a<b<c and a+b+c=3/2, then the value of a is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If in a ΔABC acos2(2C​)+ccos2(2A​)=23b​, then the sides a,b and c

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the positive numbers a,b,c,d be in A.P. Then abc,abd,acd,bcd are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the system of linear equations x+2ay+az=0; x+3by+bz=0; x+4cy+cz=0; has a non-zero solution, then a,b,c

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The ex-radii r1​,r2​,r3​ of ΔABC are in H.P. Show that its sides a,b,c are in A.P.

If $a, b, c$ are in A.P., $a^{2}, b^{2}, c^{2}$ are in H.P., then prove that either $a = b = c$ or $a, b, - c /2$ form a G.P.

Substituting $b$ into the H.P. Equation

Case 1: $a = b = c$

If the first and the $(2 n - 1)$ st terms of an A.P., a G.P. and an H.P. are equal and their $n$ th terms are $a, b$ and $c$ respectively, then

If $ln (a + c), ln (a - c), ln (a - 2 b + c)$ are in A.P., then

If $x = \sum_{n = 0}^{\infty} a^{n}, y = \sum_{n = 0}^{\infty} b^{n}, z = \sum_{n = 0}^{\infty} c^{n}$ where $a, b, c$ are in A.P and $∣ a ∣ < 1, ∣ b ∣ < 1, ∣ c ∣ < 1$ then $x, y, z$ are in

If $a, b, c$ are in G.P., then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root if $d / a, e / b, f / c$ are in

Suppose $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. if $a < b < c$ and $a + b + c = 3/2$ , then the value of $a$ is

If in a $Δ A B C$ $a cos^{2} (\frac{C}{2}) + c cos^{2} (\frac{A}{2}) = \frac{3 b}{2}$ , then the sides $a, b$ and $c$

Let the positive numbers $a, b, c, d$ be in A.P. Then $ab c, ab d, a c d, b c d$ are

If the system of linear equations $x + 2 a y + a z = 0$ ; $x + 3 b y + b z = 0$ ; $x + 4 cy + cz = 0$ ; has a non-zero solution, then $a, b, c$

The ex-radii $r_{1}, r_{2}, r_{3}$ of $Δ A B C$ are in H.P. Show that its sides $a, b, c$ are in A.P.

If $a, b, c$ are in A.P., $a^{2}, b^{2}, c^{2}$ are in H.P., then prove that either $a = b = c$ or $a, b, - c /2$ form a G.P.

If the first and the $(2 n - 1)$ st terms of an A.P., a G.P. and an H.P. are equal and their $n$ th terms are $a, b$ and $c$ respectively, then

If $ln (a + c), ln (a - c), ln (a - 2 b + c)$ are in A.P., then

If $x = \sum_{n = 0}^{\infty} a^{n}, y = \sum_{n = 0}^{\infty} b^{n}, z = \sum_{n = 0}^{\infty} c^{n}$ where $a, b, c$ are in A.P and $∣ a ∣ < 1, ∣ b ∣ < 1, ∣ c ∣ < 1$ then $x, y, z$ are in

If $a, b, c$ are in G.P., then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root if $d / a, e / b, f / c$ are in

Suppose $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. if $a < b < c$ and $a + b + c = 3/2$ , then the value of $a$ is

If in a $Δ A B C$ $a cos^{2} (\frac{C}{2}) + c cos^{2} (\frac{A}{2}) = \frac{3 b}{2}$ , then the sides $a, b$ and $c$

Let the positive numbers $a, b, c, d$ be in A.P. Then $ab c, ab d, a c d, b c d$ are

If the system of linear equations $x + 2 a y + a z = 0$ ; $x + 3 b y + b z = 0$ ; $x + 4 cy + cz = 0$ ; has a non-zero solution, then $a, b, c$

The ex-radii $r_{1}, r_{2}, r_{3}$ of $Δ A B C$ are in H.P. Show that its sides $a, b, c$ are in A.P.