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Byju's Answer

Standard XII

Mathematics

Sum of n Terms

A1, A2, A3, ....

Question

$A_{1}, A_{2}, A_{3}, . . . A_{n}$ are $n$ points in a plane whose coordinates are $(x_{1}, y_{1})$ , $(x_{2}, y_
{2}) $,$ (x_{3}, y_{3}) $. . .,$ (x_{n}, y_{n})$ respectively.

If $x_{1} = a, y_{2} = b; x_{1}, x_{2}, . . . x_{n}$ and $y_{1}, y_{2}, . . . y_{n}$ form an ascending arithmetic progression with common difference $2$ and $4$ respectively, then the coordinates of $G$ in $Q 1$ are

(a + n - 1, b + 2 (n - 1))

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(a + 2 n - 1, b + n - 1)

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(a + n - 1, b + n + 1)

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(a + n - 1, b + n + 2)

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Solution

The correct option is A $(a + n - 1, b + 2 (n - 1))$
Now if $x_{1} = a, y_{1} = b$
then $x_{2} = a + 2, x_{3} = a + 4 . . . x_{n} = a + (n - 1) 2$
$y_{2} = b + 4, y_{3} = b + 8 . . . y_{n} = b + (n - 1) 4$
and the coordinates of $G$ are
$(\frac{1}{n} \times \frac{n}{2} [a + a + (n - 1) 2], \frac{1}{n} \times \frac{n}{2} [b + b + (n - 1) 4])$
$= (a + n - 1, b + 2 (n - 1))$

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Similar questions

Q. A, B, C, D... are n points in a plane whose coordinates are

(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}), . . . . . . A B

is bisected in the point

G_{1}; G_{2} C

is divided at

G_{3}

in the ratio

1 : 2; G_{3} D

is divided at

G_{4}

in the ratio

1 : 3; G_{4} E

G_{5}

in the ratio 1 : 4, and so on until all the points are exhausted. Show that the coordinates of the final point so obtained are

\frac{x_{1} + x_{2} + x_{3} + . . . . . + x_{n}}{n}

and

\frac{y_{1} + y_{2} + y_{3} + . . . + y_{n}}{n}

[This point is called the Centre of Mean Position of the n given points.]

If $x_{1}, x_{2}, . . . ., x_{n}$ are n values of a variable X and $y_{1}, y_{2}, . . . . y_{n}$ are n values of variable Y such that $y_{i} = a x_{i} + b, i = 1, . . . ., n .$ then write Var(Y) interms of Var (X).

Q. lf

p, x_{1}, x_{2}, x_{3} \dots \dots \dots

and

q, y_{1}, y_{2}, y_{3} \dots \dots

from two infinite AP's with common difference

a

and

b

respectively, then the locus of

P (α, β)

where

α = \frac{x_{1} + x_{2} \dots \dots . + x_{n}}{n}

β = \frac{y_{1} + y_{2} \dots \dots . + y_{n}}{n}

Q. If the standard deviation of

n

observations

x_{1}, x_{2}, . . ., x_{n}

4

and another set of

n

observations

y_{1}, y_{2}, . . ., y_{n}

3

. The standard deviation of

n

observations

x_{1} - y_{1}, x_{2} - y_{2}, . . ., x_{n} - y_{n}

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

are

n

points in a plane whose coordinates are

(x_{1}, y_{1})

, $(x_{2}, y_

{2})

,

(x_{3}, y_{3})

. . .,

(x_{n}, y_{n})$ respectively.

P_{1}

is the centroid of the triangle

A_{1}, A_{2}, A_{3}, P_{2}

is the centroid of the triangle

A_{2}, A_{3}, A_{4}, P_{3}

is that of the triangle

A_{3}, A_{4}, A_{5}

and so on

P_{n - 2}

is the centroid
of the points

A_{n - 2}, A_{n - 1}, A_{n}

. The coordinates of the centre of mean position of

P_{1}, P_{2}, . . . P_{n - 2}

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A1,A2,A3,...An are n points in a plane whose coordinates are (x1,y1), $(x_{2}, y_{2}),(x_{3}, y_{3})...,(x_{n}, y_{n})$ respectively.If x1=a,y2=b;x1,x2,...xn and y1,y2,...yn form an ascending arithmetic progression with common difference 2 and 4 respectively, then the coordinates of G in Q1 are

The correct option is A (a+n−1,b+2(n−1))Now if x1=a,y1=bthen x2=a+2,x3=a+4...xn=a+(n−1)2 y2=b+4,y3=b+8...yn=b+(n−1)4and the coordinates of G are(1n×n2[a+a+(n−1)2],1n×n2[b+b+(n−1)4])=(a+n−1,b+2(n−1))