Question

Question 8
Sum of an infinitely long G.P. is 125. If 32 is added to the second term of the series,
then the first three terms form an A.P. Find the common difference of the A.P. thus
formed.​

Answer 1

Given:

Co - ordinates of point A = (-1 , 7)

Co - ordinates of point B = (4, - 3)

Ratio in which P divides A and B is 2:3.

⠀⠀⠀⠀⠀⠀⠀

To find:

Co - ordinates of point P ?

⠀⠀⠀⠀⠀⠀⠀

Solution:

⠀⠀⠀⠀⠀⠀⠀

☯ Let coordinates of point P be (x,y).

⠀⠀⠀⠀⠀⠀⠀

$\dag\;{\underline{\frak{Using\;section\;formula\;:}}}$

⠀⠀⠀⠀⠀⠀⠀

$\star\;{\boxed{\sf{\purple{(x,y) = \bigg( \dfrac{m_2 x_1 + m_1 x_2}{m_1 + m_2}\;,\; \dfrac{m_2 y_1 + m_1 y_2}{m_1 + m_2} \bigg)}}}}$

$\dag\;{\underline{\frak{Putting\;values,}}}$

⠀⠀⠀⠀⠀⠀⠀

$:\implies\sf (x,y) = \bigg( \dfrac{3 \times (-1) + 2 \times 4}{2 + 3}\;,\; \dfrac{3 \times (7) + 2 \times (-3)}{2 + 3} \bigg)$

$:\implies\sf (x,y) = \bigg( \dfrac{8 - 3}{5}\;,\; \dfrac{-6 + 21}{5}\bigg)$

$:\implies\sf (x,y) = \bigg( \dfrac{5}{5}\;,\; \dfrac{15}{5}\bigg)$

$:\implies\sf \pink{(x,y) = (1 , 3)}\;\bigstar$

$\therefore\;{\underline{\sf{Thus,\; Coordinates\;of\;point\;P\;are\; {\textsf{\textbf{(1,3)}}}.}}}$

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━

$\qquad\qquad\boxed{\bf{\purple{\mid{\overline{\underline{\bigstar\:More\;to\;know:}}}}\mid}}$

Formula that is used to find the distance between two points :

⠀⠀⠀⠀⠀⠀⠀

Distance Formula = $\sf \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Answer 2

Answer:

-48

Step-by-step explanation:

Sum of infinitely long GP = a/(1-r)  Here, given as 125.

so, a/(1-r) = 125

a= 125(1-r).    ---- (1)

now, we know for GP sequence is = a ,ar,a$r^{2}$...

if 32 is added to second term then it becomes (ar+32)

now,   a, ar+32, a$r^{2}$ form an AP.

in AP,  d = (a+d) - a = (a+2d)-(a+d)

thus here,

ar+32-a = a$r^{2}$ - (ar+32)

ar+32-a=ar^2-ar-32

a$r^{2}$-2ar+a = 32+32

a(r^2 -2r +1) = 64

$(1-r)^{2}$a =64

inserting value of "a" from equation (1),

125(1-r)^3 = 64

(1-r)^3 = 64/125

1-r = $\sqrt[3]{64/125}$

1-r = 4/5

r = 1/5.

and, a= 125(1- (1/5))

a= 100.

making second term = 100*(1/5) + 32 = 52

therefore, common difference, d = 52 - 100 = -48.