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Arithmetic Mean

ABCD is a squ...

Question

$A B C D$ is a square of length $a, a \in N, a > 1$ . Let $L_{1}, L_{2}, L_{3} . . . . . .,$ be point on BC such that $B L_{1} = L_{1} L_{2} = L_{2} L_{3} = . . . . . = 1$ and $M_{1}, M_{2}, M_{3} . . . . . . . . .$ be points on CD such that $C M_{1} = M_{1} M_{2} = . . . . . . . . . . . = 1$ . Then $a - 1 \sum n = 1 (A L_{2}^{2} + L_{n} M_{n}^{2})$ is equal to

A

\frac{1}{2} a (a - 1)^{2}

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B

\frac{1}{2} (a - 1) (2 a - 1) (4 a - 1)

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C

\frac{1}{2} a (a - 1) (4 a - 1)

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D

None of these

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Solution

The correct option is C $\frac{1}{2} a (a - 1) (4 a - 1)$
${(A L_{n})}^{2} = {(A B)}^{2} + {(B L_{n})}^{2} = a^{2} + n^{2} {(L_{n} M_{n})}^{2} = {(a - n)}^{2} + n^{2} a - 1 \sum n = 1 ({(A L_{n})}^{2} + {(L_{n} M_{n})}^{2}) = a - 1 \sum n = 1 (a^{2} + n^{2} + n^{2} + n^{2} - 2 a n + a^{2}) a - 1 \sum n = 1 (3 n^{2} - 2 a n + 2 a^{2}) = 3 a - 1 \sum n = 1 n^{2} - 2 a a - 1 \sum n = 1 n + 2 a^{2} a - 1 \sum n = 1 (1) \frac{3 (a - 1) a (2 a - 1)}{6} - \frac{2 a (a - 1) a}{2} + 2 a^{2} (a - 1) (a - 1) a [\frac{(2 a - 1)}{2} - a + 2 a] = \frac{1}{2} a (a - 1) (4 a - 1)$

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Similar questions

A B C D

is a square of length

a, a \in N, a > 1.

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

B C

B L_{1} = L_{1} L_{2} = L_{2} L_{3} = \dots = 1

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

C D

C M_{1} = M_{1} M_{2} = M_{2} M_{3} = \dots = 1.

a - 1 \sum n = 1 (A {L_{n}}^{2} + L_{n} {M_{n}}^{2})

Q. ABCD is a square of length a,

a ϵ N

L_{1}

L_{2}

L_{3}

,... be points on BC such that

B L_{1} = L_{1} L_{2} = L_{2} L_{3} = . . . = 1

M_{1}

M_{2}

M_{3}

,... be points on CD such that

C M_{1} = M_{1} M_{2} = M_{2} M_{3} = . . . = 1

\sum_{n = 1}^{a - 1} (A {L_{n}}^{2} + L_{n} {M_{n}}^{2})

Q. PQRS is a square of length a, a natural number >1. Let

L_{1}, L_{2}, L_{3}, L_{4} \dots

be points on QR such that Q

L_{1} = L_{1} L_{2} = L_{2} L_{3} = L_{3} L_{4} \dots . = 1

M_{1}, M_{2}, M_{3}

be points on RS such that R

M_{1} = M_{1} M_{2} = M_{2} M_{3} . . . .

_{n = 1}^{a - 1} \sum (P L_{n}^{2} + L_{n} M_{n}^{2})

A B C D

is a square of length

a, a \in N, a > 1.

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

B C

B L_{1} = L_{1} L_{2} = L_{2} L_{3} = \dots = 1

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

C D

C M_{1} = M_{1} M_{2} = M_{2} M_{3} = \dots = 1.

a - 1 \sum n = 1 (A {L_{n}}^{2} + L_{n} {M_{n}}^{2})

A B C

is a right angled

△

∠ B = 90^{\circ}

B C = a

n

L_{1}, L_{2}, L_{3}, . . ., L_{n}

A B

A B

is divided into

(n + 1)

equal parts and

L_{1} M_{1}, L_{2} M_{2}, . . . L_{n} M_{n}

are line segments parallel to

B C

M_{1}, M_{2}, M_{3}, . . ., M_{n}

A C

, then the sum of the lengths of

L_{1} M_{1}, L_{2} M_{2}, . . . L_{n} M_{n}

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