Question

Consider two different infinite geometric progressions with their sums S1 and S2 as
S1 = a + ar + ar² + ar³ + ..........
S2 = b + bR + bR² + bR³ + ........
If S1 = S2 = 1, ar = bR and ar²=1/8
The sum of their common ratios is::?????​

Answer 1

Answer: 1

Step-by-step explanation:

1) We are given two different infinite geometric progressions with their sums as given below.

$S_1=a+ar+ar^2+ar^3+....\infty \\ \\S_2=b+bR+bR^2+.....\infty$

Here, according to the question

ar = bR .

Since, both series are different

$=>R\neq r$

2) Since, both GP's are finite.

$1=S_1 = \frac{a}{1-r}\\ \\ =>1 =\frac{a}{1-r}\\ \\=>a=1-r$

Similarly,

$b=1-R$

3) Since,

$ar=bR\\ \\=> (1-r)r=(1-R)R\\ \\=>r-r^2=R-R^2\\ \\=>(R-r) -(R^2-r^2)=0\\ \\=> (R-r) -(R-r) (R+r)=0 \\ \\=>(R-r) (1-(R+r))=0\\ \\=>1-(R+r)=0\\ \\ => R+r =1$

We cancelled (R-r) =0 since

$R\neq r$

Hence, sum of common ratio's is 1 .