Permutations and Combinations

Question

Find the sum of the indicated terms of each of the following CP’s:

$square root of 7 comma space square root of 21 comma space 3 square root of 7 comma space........ semicolon space space straight n space terms$

Answer

The given G.P. is: $square root of 7 comma space square root of 21 comma space 3 square root of 7 comma space.........$
Here, $straight a equals space square root of 7 comma space space straight r space equals space fraction numerator square root of 21 over denominator square root of 7 end fraction space equals space fraction numerator square root of 3 square root of 7 over denominator square root of 7 end fraction space equals space square root of 3 greater than 1$
$straight S subscript straight n space equals space fraction numerator straight a left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction space equals space fraction numerator square root of 7 left square bracket left parenthesis square root of 3 right parenthesis to the power of straight n minus 1 right square bracket over denominator square root of 3 minus 1 end fraction$