Arithmetic and Geometric Progressions

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Q1. What is the common ratio of the geometric sequence 3, 6, 12, ...?

Solution:

The common ratio is found by dividing the second term by the first: 6/3 = 2.

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Q2. What is the sum of the first 5 terms of the arithmetic sequence 2, 5, 8, ...?

Solution:

The first term a = 2, common difference d = 3. Sum = n/2 * (2a + (n-1)d) = 5/2 * (4 + 12) = 30.

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Q3. What is the value of x in the inequality 2x + 1 > 7?

Solution:

Subtract 1 from both sides: 2x > 6. Then divide by 2: x > 3.

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Q4. What is the solution to the equation 4x - 2 = 10?

Solution:

Add 2 to both sides: 4x = 12. Then divide by 4: x = 3.

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Q5. What is the solution to the inequality 2x + 1 > 5?

Solution:

Subtract 1 from both sides: 2x > 4. Then divide by 2: x > 2.

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Q6. Which expression represents the polynomial 3x^2 + 5x - 2 factored?

Solution:

To factor, find two numbers that multiply to -6 and add to 5: (3x - 1)(x + 2).

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Q7. Which expression represents the factored form of x^2 + 7x + 10?

Solution:

The expression factors to (x + 5)(x + 2) since 5 and 2 add to 7 and multiply to 10.

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Q8. What is the value of x in the equation 5(x - 1) = 20?

Solution:

Divide both sides by 5: x - 1 = 4. Then add 1: x = 5.