PQRS is a square of length a- a natural number -1- Let L 1- L 2- L 3- L 4- be points on QR such that QL 1- L 1 L 2- L 2 L 3- L 3 L 4- -1 andM 1- M 2- M 3 be points on RS such that RM 1- M 1 M 2- M 2 M 3-1- Then- n-1 a -1- P L n 2- L n M n 2 is equal to-A- 1 - 2 - a - a -12B- 1 - 2 xa-1 x2 a-1 x4 a-1C- 1 - 2 - x x a -1 x 4 a -1D- 1 - 2 xa-1 x2 a-1 x4 a-1

The correct option is C ½ x a x(a 1)x(4a 1)option (b) Assume a square of side 2 as follows with L1 as the midpoint of side QR and M1 as the mid point of side R ...

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PQRS is a squ...

Question

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 $L_{1} = L_{1} L_{2} = L_{2} L_{3} = L_{3} L_{4} \dots . = 1$ and $M_{1}, M_{2}, M_{3}$ be points on RS such that R $M_{1} = M_{1} M_{2} = M_{2} M_{3} . . . .$ . . .=1. Then, $_{n = 1}^{a - 1} \sum (P L_{n}^{2} + L_{n} M_{n}^{2})$ is equal to?

½ x a x (a-1)²

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½ x a x(a-1)x(4a-1)

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½ x(a-1)x(2a-1)x(4a-1)

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½ x(a+1)x(2a-1)x(4a-1)

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Solution

The correct option is B ½ x a x(a-1)x(4a-1)
option (b)

Assume a square of side 2 as follows with L₁ as the midpoint of side QR and M₁as the mid-point of side RS

At n=1, the expression yields a value 5+2=7

Look in the answer options for 7, when n=1. Eliminate those answer options where this is not obtained

$\frac{1}{2} \times a \times (a - 1)^{2} = 1$

$\frac{1}{2}$ $\times$ a $\times$ (a-1) $\times$ (4a-1)= 7

$\frac{1}{2}$ $\times$ (a-1) $\times$ (2a-1) $\times$ (4a-1)= $\frac{21}{2}$

$\frac{1}{2}$ $\times$ (a+1) $\times$ (2a-1) $\times$ (4a-1)= $\frac{63}{2}$

a $^{2}$ $\times$ 2n $^{2}$ = 8

answer is option (b)

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PQRS is a square of length a, a natural number >1. Let L1,L2,L3,L4…be points on QR such that QL1=L1L2=L2L3=L3L4….=1 and M1,M2,M3 be points on RS such that RM1=M1M2=M2M3..... . .=1. Then, a−1n=1∑(PL2n+LnM2n) is equal to?