Sequence and Series Questions

MathematicsSequence and Series94 Questions Found

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 2007

Comprehension Passage

Let

A_{1}, G_{1}, H_{1}

denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For

n \geq 2

, Let

A_{n - 1}

and

H_{n - 1}

have arithmetic, geometric and harmonic means as

A_{n}, G_{n}, H_{n}

respectively.

Question 1:

Which one of the following statements is correct ?

(A)

G_{1} > G_{2} > G_{3} > \dots

(B)

G_{1} < G_{2} < G_{3} < \dots

(C)

G_{1} = G_{2} = G_{3} = \dots

(D)

G_{1} < G_{3} < G_{5} < \dots

and

G_{2} > G_{4} > G_{6} > \dots

Answer:C

Question 2:

Which one of the following statements is correct ?

(A)

A_{1} > A_{2} > A_{3} > \dots

(B)

A_{1} < A_{2} < A_{3} < \dots

(C)

A_{1} > A_{2} > A_{5} > \dots

and

A_{2} < A_{4} < A_{6} < \dots

(D)

A_{1} < A_{3} < A_{5} < \dots

and

A_{2} > A_{4} > A_{6} > \dots

Answer:A

Question 3:

Which one of the following statements is correct?

(A)

H_{1} > H_{2} > H_{3} > \dots

(B)

H_{1} < H_{2} < H_{3} < \dots

(C)

H_{1} > H_{3} > H_{5} > \dots

and

H_{2} < H_{4} < H_{6} < \dots

(D)

H_{1} < H_{3} < H_{5} < \dots

and

H_{2} > H_{4} > H_{6} > \dots

Answer:B

MathematicsArithmetic Progression (A.P.)JEE Advanced 2007

View in:English Hindi

Comprehension Passage

Let

V_{r}

denote the sum of first

r

terms of an arithmetic progression (A.P.) whose first term is

r

and the common difference is

(2 r - 1)

. Let

T_{r} = V_{r + 1} - V_{r} - 2

and

Q_{r} = T_{r + 1} - T_{r}

for

r = 1, 2, \dots

Question 1:

The sum $V_{1} + V_{2} + \dots + V_{n}$ is

(A)

\frac{1}{12} n (n + 1) (3 n^{2} - n + 1)

(B)

\frac{1}{12} n (n + 1) (3 n^{2} + n + 2)

(C)

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

(D)

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

Answer:B

Question 2:

$T_{r}$ is always

(A)

an odd number

(B)

an even number

(C)

a prime number

(D)

a composite number

Answer:D

Question 3:

Which one of the following is a correct statement ?

(A)

Q_{1}, Q_{2}, Q_{3}, \dots

are in A.P. with common difference 5

(B)

Q_{1}, Q_{2}, Q_{3}, \dots

are in A.P. with common difference 6

(C)

Q_{1}, Q_{2}, Q_{3}, \dots

are in A.P. with common difference 11

(D)

Q_{1} = Q_{2} = Q_{3} = \dots

Answer:B

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 1982

View in:English Hindi

$mn$ squares of equal size are arranged to from a rectangle of dimension $m$ by $n$ , where $m$ and $n$ are natural numbers. Two squares will be called 'neighbours' if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighbouring squares. Show that this is possible only if all the numbers used are equal.

MathematicsSum of Special SeriesJEE Advanced 1983

View in:English Hindi

Use mathematical Induction to prove : If $n$ is any odd positive integer, then $n (n^{2} - 1)$ is divisible by 24.

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 1984

View in:English Hindi

If $a > 0, b > 0$ and $c > 0$ , prove that $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$ .

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 1991

View in:English Hindi

Let $p$ be the first of the $n$ arithmetic means between two numbers and $q$ the first of $n$ harmonic means between the same numbers. Show that $q$ does not lie between $p$ and $(\frac{n + 1}{n - 1})^{2} p$ .

MathematicsArithmetic Progression (A.P.)JEE Advanced 2000

View in:English Hindi

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

MathematicsGeometric Progression (G.P.)JEE Advanced 2001

View in:English Hindi

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.)

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

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 $\frac{G _{1} G _{2}}{H _{1} H _{2}} = \frac{A _{1} + A _{2}}{H _{1} + H _{2}} = \frac{( 2 a + b ) ( a + 2 b )}{9 ab}$ .

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

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.

...10

Comprehension Passage

Which one of the following statements is correct ?

Which one of the following statements is correct ?

Which one of the following statements is correct?

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum V1​+V2​+⋯+Vn​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tr​ is always

Which one of the following is a correct statement ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Use mathematical Induction to prove : If n is any odd positive integer, then n(n2−1) is divisible by 24.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a>0,b>0 and c>0, prove that (a+b+c)(a1​+b1​+c1​)≥9.

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

.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.

Comprehension Passage

Which one of the following statements is correct ?

Which one of the following statements is correct ?

Which one of the following statements is correct?

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum V1​+V2​+⋯+Vn​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tr​ is always

Which one of the following is a correct statement ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Use mathematical Induction to prove : If n is any odd positive integer, then n(n2−1) is divisible by 24.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a>0,b>0 and c>0, prove that (a+b+c)(a1​+b1​+c1​)≥9.

.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.

The sum $V_{1} + V_{2} + \dots + V_{n}$ is

$T_{r}$ is always

Use mathematical Induction to prove : If $n$ is any odd positive integer, then $n (n^{2} - 1)$ is divisible by 24.

If $a > 0, b > 0$ and $c > 0$ , prove that $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$ .

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.

The sum $V_{1} + V_{2} + \dots + V_{n}$ is

$T_{r}$ is always

Use mathematical Induction to prove : If $n$ is any odd positive integer, then $n (n^{2} - 1)$ is divisible by 24.

If $a > 0, b > 0$ and $c > 0$ , prove that $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$ .

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.