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Maths-

General

Easy

Question

Let $P equals 3 to the power of 1 divided by 3 end exponent times 9 to the power of 1 divided by 9 end exponent times 27 to the power of 1 divided by 27 end exponent horizontal ellipsis horizontal ellipsis infinity$ then $P to the power of 1 divided by 3 end exponent equals$

$3^{2 / 3}$
$\sqrt{3}$
$3^{1 / 3}$
$3^{1 / 4}$

hint Hint:

In GP Sum of infinite terms: S∞ = a / (1 - r) when |r| < 1 and the sum is NOT defined when |r| ≥ 1.

The correct answer is: $3 to the power of 1 divided by 4 end exponent$

Given : $P equals 3 to the power of 1 divided by 3 end exponent times 9 to the power of 1 divided by 9 end exponent times 27 to the power of 1 divided by 27 end exponent horizontal ellipsis horizontal ellipsis infinity$
To Find : $P to the power of 1 divided by 3 end exponent equals$ ?
Detailed Solution
$P equals 3 to the power of 1 divided by 3 end exponent times 9 to the power of 1 divided by 9 end exponent times 27 to the power of 1 divided by 27 end exponent horizontal ellipsis horizontal ellipsis infinity C a n space a l s o space b e space w r i t t e n space a s P equals 3 to the power of 1 divided by 3 end exponent times 3 to the power of 2 divided by 9 end exponent times 3 to the power of 3 divided by 27 end exponent horizontal ellipsis horizontal ellipsis infinity rightwards double arrow P space equals space 3 to the power of open parentheses 1 third space plus space 2 over 9 space plus space 3 over 27 space..... infinity space close parentheses end exponent space............... space left parenthesis space x to the power of m cross times space x to the power of n space equals space space x to the power of m space plus space n end exponent right parenthesis L e t space S space equals space 1 third space plus space 2 over 3 squared space plus space 3 over 3 cubed space..... infinity space.............. space left parenthesis 1 right parenthesis D i v i d i n g space b y space 3 space o n space b o t h space s i d e s rightwards double arrow S over 3 space equals space 1 over 3 squared space plus space 2 over 3 cubed space plus space 3 over 3 to the power of 4 space.... infinity space space.................. space left parenthesis 2 right parenthesis O n space s u b t r a c t i n g space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space g e t rightwards double arrow S open parentheses 1 minus 1 third close parentheses space space equals space 1 third space plus space 1 over 3 squared space left parenthesis 2 minus 1 right parenthesis space plus space 1 over 3 cubed left parenthesis 3 minus 2 right parenthesis plus space........... space infinity space rightwards double arrow S open parentheses 2 over 3 close parentheses space equals space 1 third plus space 1 over 3 squared plus space 1 over 3 cubed space plus space.......... infinity space rightwards double arrow 2 S space equals 3 space left parenthesis 1 third plus space 1 over 3 squared plus space 1 over 3 cubed space plus space.......... infinity space right parenthesis rightwards double arrow 2 S space equals 1 plus space 1 third plus space 1 over 3 squared space plus space.......... infinity space rightwards double arrow S space equals space 1 half left parenthesis 1 plus space 1 third plus space 1 over 3 squared space plus space.......... infinity right parenthesis T h e space a b o v e space s e r i e s space i s space i n space G P w h e r e space a space equals space 1 space a n d space r space equals 1 third I n space G P space S u m space o f space i n f i n i t e space t e r m s colon space S infinity space equals space a space divided by space left parenthesis 1 space minus space r right parenthesis space w h e n space vertical line r vertical line space less than space 1 space a n d space t h e space s u m space i s space N O T space d e f i n e d space w h e n space vertical line r vertical line space greater or equal than space 1. rightwards double arrow S space equals space 1 half space open square brackets fraction numerator 1 over denominator 1 minus begin display style 1 third end style end fraction close square brackets space equals space 1 half space open square brackets fraction numerator 1 over denominator begin display style 2 over 3 end style end fraction close square brackets space equals space space 3 over 4 rightwards double arrow space P space equals space 3 to the power of 3 over 4 end exponent N o w space P to the power of 1 third end exponent space equals space open parentheses 3 to the power of 3 over 4 end exponent close parentheses to the power of 1 third end exponent space equals space open parentheses 3 to the power of 1 fourth end exponent close parentheses$

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