If A=1+ra+r2a+r3a+........∞ and B=1+rb+r2b+r3b+........∞ then (a/b) is equal to

Question

If A=1+ra+r2a+r3a+........∞ and B=1+rb+r2b+r3b+........∞ then (a/b) is equal to (A) log1B(1 - A) (B) log(B-1/B)((A-1/A)) (C)  logB A (D) none of these

Tardigrade Admin · Accepted Answer

A = (1/1-ra) ⇒ 1-ra = (1/A) ⇒ ra = 1-(1/A) =( A-1/A) B= (1/1-rb) ⇒ rb = 1 -(1/B) = (B-1/B) ∴ a log r = logr a = log ((A-1/A)), b log r = logab = log((B-1/B) ) ∴ (a/b) = (log((A-1/A))/log((B-1)B) ) = log(B-1/B) ((A-1/A))

Q. If $A = 1 + r^{a} + r^{2 a} + r^{3 a} + ........\infty$ and $B = 1 + r^{b} + r^{2 b} + r^{3 b} + ........\infty$ then $\frac{a}{b}$ is equal to

$l o g_{1_{B}} (1 - A)$

$l o g_{\frac{B - 1}{B}} (\frac{A - 1}{A})$

$l o g_{B} A$

none of these

Q. If A=1+ra+r2a+r3a+........∞ and B=1+rb+r2b+r3b+........∞ then ba​ is equal to

log1B​​(1−A)

logBB−1​​(AA−1​)

logB​A

none of these

A=1−ra1​ ⇒1−ra=A1​ ⇒ra=1−A1​=AA−1​ B=1−rb1​ ⇒rb=1−B1​=BB−1​ ∴alogr=logr​a=log(AA−1​),blogr =logab​=log(BB−1​) ∴ba​=log(BB−1​)log(AA−1​)​=logBB−1​​(AA−1​)

Q. If $A = 1 + r^{a} + r^{2 a} + r^{3 a} + ........\infty$ and $B = 1 + r^{b} + r^{2 b} + r^{3 b} + ........\infty$ then $\frac{a}{b}$ is equal to

$l o g_{1_{B}} (1 - A)$

$l o g_{\frac{B - 1}{B}} (\frac{A - 1}{A})$

$l o g_{B} A$