Which term of the Geometric Progression (GP) 5, 10, 20 40, ------is 1280?

Given:

• Geometric Progression (GP): 5, 10, 20, 40, ...

• Common ratio (^r): r = 10 / 5 = 2

• First term (^a): a = 5

• We need to find the term that equals 1280.

Concept Used:

The n^th term of a Geometric Progression (GP) is given by: \(T_n=ar^{n-1} \) 

Where:

• ^T_n: n^th term

• a: First term

• r: Common ratio

• n: Position of the term

Calculation:

We are given ^T_n = 1280.

Using the formula:

1280 = 5 × 2^n-1

Divide both sides by 5:

⇒ 256 = 2^n-1

Express 256 as a power of 2:

⇒ 256 = 2⁸

Equating the exponents:

⇒ n - 1 = 8

Solving for n:

⇒ n = 8 + 1

⇒ n = 9

Conclusion:

∴ The 9^th term of the GP is 1280.

Final Answer:

Option 1: 9^th