Question

A rope is cut into p parts. The length of each part increases and forms a geometric
progression. The length of the seventh part of the rope is 9 times the length of the fifth part of the rope.
Find the common ratio.
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Answer 1

Step-by-step explanation:

Given :-

A rope is cut into p parts. The length of each part increases and forms a geometric progression. The length of the seventh part of the rope is 9 times the length of the fifth part of the rope.

To find :-

Find the common ratio ?

Solution :-

Given that

Number of parts a rope is cut = p parts

The length of each part increases and forms a geometric progression.

Let they be : a , ar , ar² , ... ,ar^(p-1)

We know that

a = First term

r = Common ratio

nth term of the GP = an = a×r^(n-1)

The length of the fifth part = a5

=>a×r^(5-1)

=> ar⁴

a5 = ar⁴

The length of the seventh part =a7

=> a× r^(7-1)

=> ar⁶

a7 = ar⁶

Given that

The length of the seventh part = 9 times the length of the fifth part of the rope.

=> a7 = 9×a5

=> ar⁶ = 9×ar⁴

=> ar⁶/ar⁴ = 9

=> r⁶/r⁴ = 9

=> r^(6-4) = 9

Since a^m / a^n = a^(m-n)

=> r² = 9

=> r =±√9

=> r = ±3

Since the length of each part increases then GP is in increasing order.

So, r = 3

Answer:-

The value of the common ratio for the given problem is 3

Used formulae:-

• nth term of the GP = an = a×r^(n-1)

Where,

• a = First term

• r = Common ratio

• n = Number of terms

• a^m / a^n = a^(m-n)