General term of GP
Nth term of a GP is given by an = a r n−1 Thus , a , G . P . can be written as a , ar , ar 2 , ar 3 , … ar n – 1 . The series a + ar + ar 2 + ... + ar n – 1 is called finite geometric series
The series a + ar + ar 2 + ... + ar n – 1 +................ is called infinite geometric series
Numerical : In a G . P ., the 3rd term is 24 and the 6th term is 192 . Find the 10th term
Solution : Nth term of a GP is given by an = a r n−1 3 rd term , a3 = ar 2 = 24 --- ( i ) 6 th term , a6 = ar 5 = 192 ---- ( ii ) Dividing ( ii by ( i ), we get r 3 = 8 or r = 2 . Substituting r = 2 in ( i ), we get a = 6 . Hence a10 = 6 ( 2 ) 9 = 3072
Sum of terms of a GP :
Sn = a ( 1-r n ) /( 1-r ) or Sn = a ( r n -1 ) /( r-1 ) Numerical : Find sum of first 5 terms of geometric series 1 , 2 / 3 , 4 / 9 …. Solution : in this case a = 1 , r = 2 / 3
Sn = a ( 1-r n ) /( 1-r )
S5 = a ( 1-r 5 ) /( 1-r ) Or S5 = 1 ( 1 – ( 2 / 3 ) 5 )/( 1-2 / 3 ) = 211 / 81