Question

the product of 10 GM 's between 1 and 49

Answer 1

Given : product of 10 GMs between 1and 49​ is p .  & GM of 1 & 49 is G

To find : Show that p = G¹⁰

Solution:

10 GMs inserted between 1 & 49

Hence total terms =  12

12th term = a r¹¹

49  =  1 * r¹¹

=> r¹¹  = 49

10 GMs

1*r  ,  1*r² , ................., 1*r¹⁰

=  r  ,   r² , .................,  r¹⁰

Product of 10 GMs = r *  r²* ........*  r¹⁰

= r⁽¹⁺²⁺       ⁺¹⁰⁾

1 + 2 + 3 + ........+ 10  = 10 * 11/2 =  5 * 11  = 55

= r⁵⁵

= (r¹¹)⁵

= (49)⁵

Product of 10 GMs = p = (49)⁵

GM of 1 & 49  = √1 * 49  =  √49  = G

p  = (49)⁵  

=> p = (( √49)² )⁵

=> p = (G²)⁵

=> p =  G¹⁰

QED

Hence Proved

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