Question

How many terms of the series `2/3 - 1 + 3/2` .... has the sum `463/96`?

Answer 1

a = `2/3`

r = `(-1)/(2/3)`

= `(-3)/2 < 1`

`S_n = 463/96`

`Sn = (a(1 - r^n))/(1 - r)`

`463/96 = (2/3 (1 - ((-3)/2)^n))/(1 - ((-3)/2))`

`463/(32 xx 2) = (1 - ((-3)/2)^n) xx 2/5`

`(463 xx 5)/(64 xx 2) = 1 - ((-3)/2)^n`

`((-3)/2)^n = 1 - 2315/128`

`((-3)/2)^n = (128 - 2315)/128`

`((-3)/2)^n = (-2187)/128`

`((-3)/2)^n = (-3^7)/2^7`

`((-3)/2)^n = ((-3)/2)^7`

n = 7