Question

If \( a_1, a_2, a_3, \dots \) are the terms of an increasing geometric progression such that \[ a_1 + a_3 + a_5 = 21, \quad a_1a_3a_5 = 64 \] then \[ a_1 + a_2 + a_3 \] is

• A. 5
• B. 7
• C. 10
• D. 15

Hint:

For geometric progressions, always use the ratio of terms to express terms in terms of a single variable and simplify accordingly.

Answer 1

The Correct Option is B

Step 1: Let \( a_3 = P, a_1 = \frac{P}{r^2}, a_5 = P r^2 \).
We are given the equations: \[ a_1 + a_3 + a_5 = 21 \quad \text{and} \quad a_1 a_3 a_5 = 64 \] Substitute \( a_1 = \frac{P}{r^2}, a_3 = P, a_5 = P r^2 \) into the equations: \[ \frac{P}{r^2} + P + P r^2 = 21 \] \[ \frac{P}{r^2} \cdot P \cdot P r^2 = 64 \] Step 2: Solve the equations.
From \( \frac{P}{r^2} + P + P r^2 = 21 \), multiply by \( r^2 \): \[ P + P r^2 + P r^4 = 21r^2 \] This simplifies to: \[ 1 + r^2 + r^4 = \frac{21}{4} \] Step 3: Solve for \( r \).
Solving the above equation leads to: \[ r = \pm 1 \] Step 4: Find the sum \( a_1 + a_2 + a_3 \).
Substituting the values, we find: \[ a_1 + a_2 + a_3 = 1 + 2 + 4 = 7 \]