If s be the sum p be the product

Asked by admin @ 13/08/2021 in Math viewed by 285 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 @ 13/08/2021

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

If s be the sum p be the product

Similar Questions

Let 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.

If sum of first p terms of an ap

If the sum of p terms of an ap

The product of x and y added to their sum

Product sum difference

Find a quadratic polynomial the sum and 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)

Find quadratic equation from sum and product of roots

The sum of the first number squared and the second number is 57 and the product is a maximum.

Overcoming fixed mindset: a step-by-step guide to cultivating a growth mindset

Which of the following would best complete this list?

What had the king decided to do before he saw the spider

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

Oxford new enjoying mathematics class 7 solutions chapter 1

Survey of various types of bank accounts icse project

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