Question 16

Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.

Answer

Let a be the first term and r be the common ratio of the G.P.

According to the given conditions,

$S subscript 2 space equals space minus 4 space equals space fraction numerator a open parentheses 1 minus r squared close parentheses over denominator 1 minus r end fraction space space space space space space space space space... left parenthesis 1 right parenthesis a subscript 5 space equals space 4 space cross times space a subscript 3 a r to the power of 4 space equals space 4 a r squared rightwards double arrow r squared space equals space 4 therefore space r space equals space plus-or-minus 2$

From (1), we obtain

$minus 4 space equals space fraction numerator a open square brackets 1 minus open parentheses 2 close parentheses squared close square brackets over denominator 1 minus 2 end fraction space f o r space r space equals 2 rightwards double arrow minus 4 space equals space fraction numerator a open parentheses 1 minus 4 close parentheses over denominator minus 1 end fraction rightwards double arrow minus 4 space equals space a open parentheses 3 close parentheses rightwards double arrow a space equals space fraction numerator minus 4 over denominator 3 end fraction A l s o comma space minus 4 space equals space fraction numerator a open square brackets 1 minus open parentheses minus 2 close parentheses squared close square brackets over denominator 1 minus open parentheses minus 2 close parentheses end fraction space f o r space r space equals minus 2 rightwards double arrow minus 4 space equals space fraction numerator a open parentheses 1 minus 4 close parentheses over denominator 1 plus 2 end fraction rightwards double arrow minus 4 space equals space fraction numerator a open parentheses minus 3 close parentheses over denominator 3 end fraction rightwards double arrow a space equals space 4 T h u s comma space t h e space r e q u i r e d space G. P. space i s fraction numerator minus 4 over denominator 3 end fraction comma space fraction numerator minus 8 over denominator 3 end fraction comma space fraction numerator minus 16 over denominator 3 end fraction space comma space... space o r space 4 comma space minus 8 comma space 16 comma space minus 32 comma space...$ .....