The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
The sum of first three terms of a G.P. is 16. Sum of the next three terms of the G.P. is 128.
Let the first term and common ratio of the given G.P. be a and r respectively.
Let the sequence of the G.P. be a , a r , a r 2 , a r 3 , …
The sum of first three terms of the G.P. is given by,
a + a r + a r 2 = 16 a ( 1 + r + r 2 ) = 16 (1)
Similarly, the sum of the next three terms of the G.P. is given by,
a r 3 + a r 4 + a r 5 = 128 a r 3 ( 1 + r + r 2 ) = 128 (2)
Divide equation (2) by equation (1).
a r 3 ( 1 + r + r 2 ) a ( 1 + r + r 2 ) = 128 16 r 3 = 8 r 3 = ( 2 ) 3 r = 2
Substitute the value of r in equation (1) to obtain a .
a ( 1 + 2 + 2 2 ) = 16 a ( 1 + 2 + 4 ) = 16 7 a = 16 a = 16 7
The formula for the sum of first n terms of the G.P. for r > 1 is given by,
S n = a ( r n − 1 ) r − 1
Substitute the values of a and r in the above expression to obtain the n t h term.
S n = 16 7 × ( 2 n − 1 ) 2 − 1 = 16 ( 2 n − 1 ) 7 S n = 16 7 ( 2 n − 1 )
Thus, the first term, common ratio and sum of n terms of the G.P. are 16 7 , 2 , 16 7 ( 2 n − 1 ) respectively.
The sum of first three terms of a G.P. is 16. Sum of the next three terms of the G.P. is 128.
Let the first term and common ratio of the given G.P. be
Let the sequence of the G.P. be
The sum of first three terms of the G.P. is given by,
Similarly, the sum of the next three terms of the G.P. is given by,
Divide equation (2) by equation (1).
Substitute the value of
The formula for the sum of first
Substitute the values of
Thus, the first term, common ratio and sum of n terms of the G.P. are
