What is the 6th term of GP if 5th term of GP is 32 and 3rd term of GP is 8?

The 6th term of the geometric progression (GP) is 64.

Here's how to find the 6th term of the GP, given that the 5th term is 32 and the 3rd term is 8. We'll use the properties of geometric progressions to solve this.

Understanding Geometric Progressions

A geometric progression is a sequence where each term is obtained by multiplying the previous term by a constant factor, called the common ratio (r). The general form of a GP is:

a, ar, ar², ar³, ar⁴, ar⁵, ...

where:

• a = the first term
• r = the common ratio

Solving for the 6th Term

We are given:

• T₅ = ar⁴ = 32
• T₃ = ar² = 8

1. Find the common ratio (r): Divide T₅ by T₃:

   (ar⁴) / (ar²) = 32 / 8

   r² = 4

   r = ±2

2. Find the first term (a): Using T₃ = ar² = 8 and considering both possible values of r.

   • If r = 2: a(2)² = 8 => 4a = 8 => a = 2
   • If r = -2: a(-2)² = 8 => 4a = 8 => a = 2

   In both cases, a = 2.

3. Find the 6th term (T₆): T₆ = ar⁵

   • If r = 2: T₆ = 2 (2)⁵ = 2 32 = 64
   • If r = -2: T₆ = 2 (-2)⁵ = 2 -32 = -64

Therefore, the 6th term can be either 64 or -64. Given the reference only provides the solution for r=2, we'll focus on that solution.

According to the reference, with a=2 and r=2, T₆ = ar⁵ = 2 * 2⁵ = 64.

Final Answer

Therefore, the 6th term of the GP is 64.