Let the first term and the common ratio of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to [2023]

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Solution

Q.
Let the first term $a$ and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to [2023]

1 231

2 241

3 210

4 220

Ans.
(1)

$We have, first term = a$

$Common ratio = r$

$According to question, a^{2} + {(a r)}^{2} + {(a r^{2})}^{2} = 33033$

$\Rightarrow a^{2} (1 + r^{2} + r^{4}) = 33033$

$\Rightarrow a = 11, r = 4$

$∴$ $Required sum = a + a r + a r^{2}$

$= 11 + 11 \times 4 + 11 {(4)}^{2} = 11 + 44 + 176 = 231$

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