What is the formula for the nth term of a geometric progression?

The formula for the nth term of a geometric progression is tₙ = a × r^(n-1), where a is the first term and r is the common ratio.

If the first term of a geometric progression is 5 and the common ratio is 3, what is the 4th term?

The 4th term t₄ = 5 × 3^(4-1) = 5 × 27 = 135.

True or False: The sum of an infinite geometric series exists only if the common ratio is greater than 1.

False. The sum exists if the absolute value of the common ratio is less than 1.

Fill in the blank: The geometric mean G between two numbers a and b is calculated as G = ___(a × b).

√

What is the sum of the first 5 terms of a geometric progression with first term 10 and common ratio 2?

The sum S₅ = 10 × (2⁵ - 1) / (2 - 1) = 10 × (32 - 1) = 10 × 31 = 310.

If the 3rd term of a geometric progression is 16 and the common ratio is 2, what is the first term?

The first term a = 16 / 2² = 16 / 4 = 4.

True or False: If a sequence is a geometric progression, the ratio of consecutive terms must be constant.

True.

Find the sum of the first 6 terms of the geometric series 1, 3, 9, 27,...

The sum S₆ = 1 × (3⁶ - 1) / (3 - 1) = 1 × (729 - 1) / 2 = 728 / 2 = 364.

If the first term of a geometric progression is 8 and the 5th term is 1, what is the common ratio?

Let r be the common ratio. Then, 8 × r⁴ = 1, so r⁴ = 1/8, thus r = 1/2.

Fill in the blank: The general form of the sum of the first n terms when the common ratio is less than 1 is Sₙ = a × (1 - rⁿ) / (1 - r).

True.

What is the 7th term of the geometric sequence where the first term is 4 and the common ratio is 3?

The 7th term t₇ = 4 × 3^(7-1) = 4 × 729 = 2916.

Which of the following statements is true regarding the properties of geometric progressions?

Multiplying terms retains properties.

• Maintains common ratio.

• Non-zero constant required.

• Sequence remains geometric.

Calculate the 10th term of the geometric progression where the first term is 2 and the common ratio is 5.

The 10th term t₁₀ = 2 × 5^(10-1) = 2 × 5⁹ = 2 × 1953125 = 3906250.

True or False: The product of the first five terms of a geometric progression is equal to the square of the 3rd term multiplied by the 4th term.

True.

The nth term of a geometric progression can be expressed as tn = ___ × ___ⁿ⁻¹.

A, r