Question

If a b c be resp. the sum of n term, and the next n term and the next n term of a GP. prove a,b,c are in GP​

Answer 1

Step-by-step explanation:

Let x be the first term & r be the common ratio of the G.P

a=

1−r

x(1−r

n

)

………(1)

b=

1−r

x(1−r

2n

)

−a

⇒a+b=

1−r

x(1−r

2n

)

…..(2)

c=

1−r

x(1−r

3n

)

−(a+b)

⇒a+b+c=

1−r

x(1−r

3n

)

……(3)

(2)÷(1)

a

a+b

=

1−r

n

1−r

2n

1+

a

b

=1+r

n

b=ar

n

(3)÷(2)

a+b

a+b+c

=

1−r

2n

1−r

3n

1+

a+b

c

=

1+r

n

1+r

n

+r

2n

a+b

c

=

1+r

n

r

2n

a+ar

n

c

=

1+r

n

r

2n

a(1+r

n

)

c

=

1+r

n

r

2n

c=ar

2n

We can say that a,ar

n

,ar

2n

are in G.P.

∴ a, b, c are in G.P.