Sequence and Series Questions

MathematicsSequence and Series94 Questions Found

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

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

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 1979Easy

The harmonic mean of two numbers is $4$ . Their arithmetic mean $A$ and the geometric mean $G$ satisfy the relation $2 A + G^{2} = 27$ . Find the two numbers.

Answer:

3 and 6 or 6 and 3

MathematicsGeometric Progression (G.P.)JEE Advanced 1982Moderate

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Does there exist a geometric progression containing $27, 8$ and $12$ as three of its terms ? If it exits, how many such progressions are possible ?

Answer:yes, infinite

MathematicsGeometric Progression (G.P.)JEE Advanced 1983Moderate

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Find three numbers $a, b, c$ , between $2$ and $18$ such that (i) their sum is $25$ (ii) the numbers $2, a, b$ sare consecutive terms of an A.P. and (iii) the numbers $b, c, 18$ are consecutive terms of a G.P.

Answer:5, 8, 12

MathematicsSum of Special SeriesJEE Advanced 1988Easy

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The sum of the first $n$ terms of the series $1^{2} + 2. 2^{2} + 3^{2} + 2. 4^{2} + 5^{2} + 2. 6^{2} + \dots$ is $n (n + 1)^{2} /2$ , when $n$ is even. When $n$ is odd, the sum is \dots.

Answer:

\frac{n ^{2} ( n + 1 )}{2}

MathematicsGeometric Progression (G.P.)JEE Advanced 1991Easy

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If $S_{1}, S_{2}, S_{3}, \dots, S_{n}$ are the sums of infinite geometric series whose first terms are $1, 2, 3, \dots, n$ and whose common ratios are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots, \frac{1}{n + 1}$ respectively, then find the values of $S_{1}^{2} + S_{2}^{2} + S_{3}^{2} + \dots + S_{2 n - 1}^{2}$ .

Answer:

\frac{n ( 2 n + 1 ) ( 4 n + 1 ) - 3}{3}

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 1992Easy

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Let the harmonic mean and geometric mean of two positive numbers be the ratio $4 : 5$ . Then the two number are in the ratio \dots.

Answer:

4 : 1 or 1 : 4

MathematicsSum of Special SeriesJEE Advanced 1996Easy

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For any odd integer $n \geq 1$ , $n^{3} - (n - 1)^{3} + \dots + (- 1)^{n - 1} 1^{3} = \dots$ .

Answer:

\frac{1}{4} (n + 1)^{2} (2 n - 1)

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 2001Moderate

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Let $a_{1}, a_{2}, \dots a_{n}$ be positive real numbers in geometric progression. For each $n$ , let $A_{n}, G_{n}, H_{n}$ be respectively, the arithmetic mean, geometric mean, and harmonic mean of $a_{1}, a_{2}, \dots a_{n}$ . Find an expression for the geometric mean of $G_{1}, G_{2}, \dots G_{n}$ in terms of $A_{1}, A_{2}, \dots A_{n}, H_{1}, H_{2}, \dots H_{n}$ .

Answer:

G = (A_{1} A_{2} \dots A_{n} H_{1} H_{2} \dots H_{n})^{1/2 n}

MathematicsArithmetic Progression (A.P.)JEE Advanced 1980Moderate

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The interior angles of a polygon are in arithmetic progression. The smallest angle is $12 0^{\circ}$ , and the common difference is $5^{\circ}$ , Find the number of sides of the polygon.

Answer:9

...10

.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; } The harmonic mean of two numbers is 4. Their arithmetic mean A and the geometric mean G satisfy the relation 2A+G2=27. Find the two numbers.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Does there exist a geometric progression containing 27,8 and 12 as three of its terms ? If it exits, how many such progressions are possible ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the first n terms of the series 12+2.22+32+2.42+52+2.62+… is n(n+1)2/2, when n is even. When n is odd, the sum is \dots.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the harmonic mean and geometric mean of two positive numbers be the ratio 4:5. Then the two number are in the ratio \dots.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For any odd integer n≥1, n3−(n−1)3+⋯+(−1)n−113=….

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The interior angles of a polygon are in arithmetic progression. The smallest angle is 120∘, and the common difference is 5∘, Find the number of sides of the polygon.

.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; } The harmonic mean of two numbers is 4. Their arithmetic mean A and the geometric mean G satisfy the relation 2A+G2=27. Find the two numbers.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Does there exist a geometric progression containing 27,8 and 12 as three of its terms ? If it exits, how many such progressions are possible ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the first n terms of the series 12+2.22+32+2.42+52+2.62+… is n(n+1)2/2, when n is even. When n is odd, the sum is \dots.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the harmonic mean and geometric mean of two positive numbers be the ratio 4:5. Then the two number are in the ratio \dots.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For any odd integer n≥1, n3−(n−1)3+⋯+(−1)n−113=….

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The interior angles of a polygon are in arithmetic progression. The smallest angle is 120∘, and the common difference is 5∘, Find the number of sides of the polygon.

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

The harmonic mean of two numbers is $4$ . Their arithmetic mean $A$ and the geometric mean $G$ satisfy the relation $2 A + G^{2} = 27$ . Find the two numbers.

Does there exist a geometric progression containing $27, 8$ and $12$ as three of its terms ? If it exits, how many such progressions are possible ?

The sum of the first $n$ terms of the series $1^{2} + 2. 2^{2} + 3^{2} + 2. 4^{2} + 5^{2} + 2. 6^{2} + \dots$ is $n (n + 1)^{2} /2$ , when $n$ is even. When $n$ is odd, the sum is \dots.

Let the harmonic mean and geometric mean of two positive numbers be the ratio $4 : 5$ . Then the two number are in the ratio \dots.

For any odd integer $n \geq 1$ , $n^{3} - (n - 1)^{3} + \dots + (- 1)^{n - 1} 1^{3} = \dots$ .

The interior angles of a polygon are in arithmetic progression. The smallest angle is $12 0^{\circ}$ , and the common difference is $5^{\circ}$ , Find the number of sides of the polygon.

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

The harmonic mean of two numbers is $4$ . Their arithmetic mean $A$ and the geometric mean $G$ satisfy the relation $2 A + G^{2} = 27$ . Find the two numbers.

Does there exist a geometric progression containing $27, 8$ and $12$ as three of its terms ? If it exits, how many such progressions are possible ?

The sum of the first $n$ terms of the series $1^{2} + 2. 2^{2} + 3^{2} + 2. 4^{2} + 5^{2} + 2. 6^{2} + \dots$ is $n (n + 1)^{2} /2$ , when $n$ is even. When $n$ is odd, the sum is \dots.

Let the harmonic mean and geometric mean of two positive numbers be the ratio $4 : 5$ . Then the two number are in the ratio \dots.

For any odd integer $n \geq 1$ , $n^{3} - (n - 1)^{3} + \dots + (- 1)^{n - 1} 1^{3} = \dots$ .

The interior angles of a polygon are in arithmetic progression. The smallest angle is $12 0^{\circ}$ , and the common difference is $5^{\circ}$ , Find the number of sides of the polygon.