Question

The fourth term of G.P is greater than first term, which is positive by 372. The third term is greater than the second by 60. Calculate the common ratio and first term of the progression.

Answer 1

let the first term be x and a be the common ratio
!st term=x
2nd term=xa
3rd term=xa2
4th term=xa3
now 
xa3-x=372
x(a3-1)=372
x=372/(a3-1)
and
xa2-xa=60
xa(a-1)=60
a(a-1)*372/(a3-1)=60
372a2-372a=60a3-60
60a3-372a2+372a-60=0
60a3-60a2-312a2+312a+60a-60=0
60a2(a-10)-312a(a-1)+60(a-1)=0
(a-1)(60a2-312a+60)=0
a=1
or
60a2-312a+60=0
10a2-52a+10=0
10a2-50a-2a+10=0
10a(a-5)-2(a-5)=0
a=5or a=2/5
but a>1
hence a=5
x*5(5-1)=60
x*20=60
x=3

First term=3 and Common ratio=5

Answer 2

Step-by-step explanation:

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

a

1

=a, a

2

=ar, a

3

=ar

2

, a

4

=ar

3

By the given condition,

a

3

=a

1

+9

⇒ar

2

=a+9....(1)

a

2

=a

4

+18

⇒ar=ar

3

+18....(2)

From (1) and (2), we get

a(r

2

−1)=9....(3)

ar(1−r

2

)=18....(4)

Dividing (4) and (3), we get

a(r

2

−1)

ar(1−r

2

)

=

9

18

⇒−r=2

⇒r=−2

Substituting the value of r in (1), we get

4a=a+9

⇒3a=9

∴a=3

⇒a

1

=3

⇒a

2

=3(−2)=−6

⇒a

3

=3(−2)

2

=12

⇒a

4

=3(−2)

3

=−24

Thus the first four numbers of the G.P are 3,-6,12 and -24.