If `S=1+a+a^2+a^3+a^4+……….to oo` and `|a|<1` then prove that `a= (S-1)/S`

If S=1+a+a^2+a^3+a^4+……….to oo then a=

If x = 1 + 3a + 6a^2 + 10 a^3 + ……" to " oo terms |a| lt 1, y = 1+4a+10a^2 + 20a^3 + ….." to " oo terms, |a| lt 1 , then x:y

If 1+2x +3x^2+4x^3 +…..oo ge 4 then

The coefficient of x^7 in (1 + 2x + 3x^2 + 4x^3 + …… "to " oo)

A line l passing through the origin is perpendicular to the lines 1: (3 + t ) hati + (-1 +2 t ) hatj + (4 + 2t) hatk -oolt t lt ooand 1__(2) : (3 + 2s )hati + (3+2s) hati + (3+ 2s) hatj + ( 2+s) hatk , -oo lt s lt oo Then the coordinate(s) of the point(s) on 1 _(2) at a distance of sqrt17 from the point of intersection of 1 and 1_(1) is (are)

If |alpha|lt1,|beta|lt1 1-alpha+alpha^(2)-alpha^(3)+ . . . . ."to "oo=s_(1) 1-beta+beta^(2)-beta^(3)+ . . . ."to "oo=s_(2) , then 1-alphabeta+alpha^(2)beta^(2)+ . . . . ."to "oo equals

If 1/1^2+1/2^2+1/3^2+...oo=pi^2/6 then value of 1-1/2^2+1/3^2-1/4^2+...oo=

If x = (1)/(1.2) + (1)/(3.4) + (1)/(5.6) …….oo, and y = 1 - (1)/(2.3) - (1)/(4.5) - (1)/(6.7)…….oo , then

y= f(x) is a polynomial function passing through point (0, 1) and which increases in the intervals (1, 2) and (3, oo) and decreases in the intervals (-oo,1) and (2, 3). If f(1)= -8, then the range of f(x) is (a) [3,oo) (b) [-8,oo) (c) [-7,oo) (d) (-oo,6]

If S_n denotes the sum of first n natural number then S_1 + S_2x + S_3 x^2 +……+S_n x^(n-1) + ……oo terms =