If the first term of a GP is 10 and the sum of the first three terms is 70, what

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Question: If the first term of a GP is 10 and the sum of the first three terms is 70, what is the common ratio?

Options:

Correct Answer: 3

Solution:

Let the common ratio be r. The sum is 10 + 10r + 10r^2 = 70. Simplifying gives r^2 + r - 6 = 0, which factors to (r - 2)(r + 3) = 0, thus r = 2.

If the first term of a GP is 10 and the sum of the first three terms is 70, what is the common ratio?

If the first term of a GP is 10 and the sum of the first three terms is 70, what is the common ratio?

Step 1: Identify the first term of the geometric progression (GP), which is given as 10.
Step 2: Let the common ratio be represented by 'r'.
Step 3: Write the first three terms of the GP: the first term is 10, the second term is 10r, and the third term is 10r^2.
Step 4: Set up the equation for the sum of the first three terms: 10 + 10r + 10r^2 = 70.
Step 5: Simplify the equation by dividing everything by 10: 1 + r + r^2 = 7.
Step 6: Rearrange the equation to form a standard quadratic equation: r^2 + r - 6 = 0.
Step 7: Factor the quadratic equation: (r - 2)(r + 3) = 0.
Step 8: Solve for r by setting each factor to zero: r - 2 = 0 or r + 3 = 0.
Step 9: Find the values of r: r = 2 or r = -3.
Step 10: Since the common ratio in a GP is usually positive, we take r = 2.

Geometric Progression (GP) – Understanding the properties of a geometric progression, including the first term, common ratio, and the formula for the sum of the first n terms.
Quadratic Equations – Solving quadratic equations through factoring or using the quadratic formula.