Question

2. If S is the sum, P is the product and R is the sum of the reciprocals of 3 terms of a G.P.,

then prove that P^2R^3/S^3
is equal 1.
​

Answer 1

Answer:

S=

r−1

a(r

n

−1)

P=a

n

×r

1+2+3+....+(n−1)

=a

n

r

2

n(n−1)

since sum of natural numbers=

2

n(n+1)

R=

a

1

+

ar

1

+

ar

2

1

+...+

ar

n−1

1

=

ar

n−1

r

n−1

+r

n−2

+r

n−3

+....+r

2

+r+1

=

r−1

1(r

n

−1)

×

ar

n−1

1

=

a(r−1)r

n−1

(r

n

−1)

∴P

2

R

n

=(a

n

r

2

n(n−1)

)

2

(

r−1

a(r

n

−1)

)

n

=

(r−1)

n

a

n

(r

n

−1)

n

=S

n

⇒P

2

R

n

=S

n

since S,P,R are in G.P(given)

P

2

=

R

n

S

n

P

2

=(

R

S

)

n

Hence proved.