Question

Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum of its squares of first three terms is 33033, then the sum of these three terms is equal to

• A. 231
• B. 220
• C. 210
• D. 241

Answer 1

The Correct Option is A

Given:

The terms \( a, ar, \) and \( ar^2 \) are in a geometric progression (GP).

• Step 1: Simplify the given equation:

\( a^2 + (ar)^2 + (ar^2)^2 = 33033 \)

\( a^2(1 + r^2 + r^4) = 33033 \)

• Factorize \( 33033 \):

\( 33033 = 11^2 \cdot 3 \cdot 7 \cdot 13 \implies a^2 = 11^2 \implies a = 11 \).

• Step 2: Substitute \( a = 11 \) into the equation:

\( 1 + r^2 + r^4 = \frac{33033}{11^2} = 3 \cdot 7 \cdot 13 = 273 \).

• Rearrange:

\( r^2(1 + r^2) = 273 - 1 = 272 \).

• Factorize \( 272 \):

\( r^2(r^2 + 1) = 16 \cdot 17 \).

• Solve for \( r^2 \):

\( r^2 = 16 \implies r = 4 \) (since \( r > 0 \)).

• Step 3: Calculate the sum of the three terms:

\( a + ar + ar^2 = a(1 + r + r^2) \).

• Substitute \( a = 11 \) and \( r = 4 \):

\( 1 + r + r^2 = 1 + 4 + 16 = 21 \).

\( a(1 + r + r^2) = 11 \cdot 21 = 231 \).

Final Answer: The sum of the three terms is \( 231 \).