Let s be the sum p be the product

Asked by admin @ 12/08/2021 in Math viewed by 260 People

Question 14 Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn

Class X1 - Maths -Sequences and Series Page 199

Answered by admin @ 12/08/2021

Let the GP is a , ar , ar² , ar³ , .......... arⁿ⁻¹.
S = Sum of n terms
= a + ar + ar² + ar³ + .......+ arⁿ⁻¹
= a( rⁿ - 1)/( r - 1)

and R = sum of reciprocal of n terms
= 1/a + 1/ar + 1/ar² + 1/ar³ + ......... + 1/arⁿ⁻¹
= 1/a[ 1 - ( 1/r )ⁿ ]/( 1 - 1/r )
= ( rⁿ - 1)r/arⁿ(r - 1)

P = product of n terms = a × ar × ar² × ar³ × .....× arⁿ⁻¹
=〈 a¹⁺¹⁺¹⁺¹⁺¹⁺¹⁺¹⁺⁻⁻¹〉〈r¹⁺²⁺³⁺⁻⁻⁻⁻⁽ⁿ⁻¹⁾〉
= aⁿ. r^{n(n-1)/2}
take square both sides,
P² = a²ⁿrⁿ⁽ⁿ⁻¹⁾

Now, we have to prove P²Rⁿ = Sⁿ
or , P² = (S/R)ⁿ

RHS = (S/R)ⁿ
= [a(rⁿ - 1)/(r - 1) ]/[(rⁿ - 1)r/arⁿ(r - 1) ]ⁿ
= [a(rⁿ - 1)/(r - 1) × arⁿ (r - 1)/(rⁿ - 1)r ]ⁿ
=〈 a²rⁿ⁻¹ 〉ⁿ
= a²ⁿ rⁿ⁽ⁿ⁻¹⁾ = P² = LHS

Let s be the sum p be the product

Similar Questions

If s be the sum p be the product

The product of a fraction and the sum of 16/3 and 19/3.what is the fraction.

Let p and q be the point of trisection

If sum of first p terms of an ap

If the sum of p terms of an ap

Let p be an interior point of triangle abc

Let p and q be the points of trisection

The product of x and y added to their sum

Product sum difference

Let p(x) = f(x)g(x) and q(x) = f(x)/g(x), where f and g are the functions whose graphs are shown.

Find a quadratic polynomial the sum and product

Navigation lets the user access relevant contents of the product

Use properties to find the sum or product of 5x23x2

A factory can produce two products x and y with a profit approximated by p=14x+22y-900

Write the sum as a product. simplify the product. (–2) + (–2) + (–2) + (–2)

Most viewed questions in Math

What is the value of x in the expression

Shubham is facing south and moves 30 km

Prachi excellence in mathematics class 7 solutions pdf free download

Survey of various types of bank accounts icse project

Oxford new enjoying mathematics class 7 solutions chapter 1

The town of p is located at point x

A clock gains 4 minutes after every 4 minutes

Running a tuck shop or canteen maths project pdf

Paul is sixteenth from the front of the row

The average weight of 20 teachers is 80 kg

The number 567 xy is completely divisible by 30

In a row of friends tiya occupies fifteenth place

What would $ mean if fat is coded as

Consider a 3 digit integer x with distinct digits

Rahul went to his mother's mother in law