Show that `(3.5)/(5.10) + (3.5.7)/(5.10.15) + (3.5.7.9)/(5.10.15.20)+ …..oo = (5sqrt5)/(3 sqrt3) - 8/5`

Show that (4.6)/(5.10) + (4.6.8)/(5.10.15) + (4.6.8.10)/(5.10.15.20) + ……. = 44/45

1-(1)/(5)+(1.4)/(5.10) - (1.5.7)/(5.10.15)+…..=

Sho that 3/6 + (3.5)/(6.9) + (3.5.7)/(6.9.12)+ ……..oo 3 sqrt3 - 4

Show that 1+3/4 + (3.5)/(4.8)+ (3.5.7)/(4.8.12)+ …… = 2sqrt2

Show that 1 - 3/4 + (3.5)/(4.8)- (3.5.7)/(4.8.12)+……..= sqrt((8/27))

If x=(1)/(5)+(1.3)/(5.10)+(1.3.5)/(5.10.15)+….oo then find 3x^(2)+6x.

If t = 4/5 + (4.6)/(5.10) + (4.6.8)/(5.10.15) + ……oo then prove that 9t = 16

solve (3sqrt5 +sqrt3)/(sqrt5 -sqrt3)

Simplify : i. (1 (5)/(7) xx (7)/(10) ) - ((3)/(5) div (9)/(10))

Given x, y in R , x^(2) + y^(2) gt 0 . Then the range of (x^(2) + y^(2))/(x^(2) + xy + 4y^(2)) is (a) ((10 - 4 sqrt(5))/(3),(10 + 4 sqrt(5))/(3)) (b) ((10 - 4 sqrt(5))/(15),(10 + 4 sqrt(5))/(15)) (c) ((5- 4 sqrt(5))/(15),(5 + 4 sqrt(5))/(15)) (d) ((20- 4 sqrt(5))/(15),(20 + 4 sqrt(5))/(15))