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Question
Find the sum of the following geometric series:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
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Solution
The given geometric series is:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
Step 1: Identify the first term (a)
a = `sqrt7`
Step 2: Find the common ratio (r)
`r = (sqrt21)/(sqrt7) = sqrt3`
Check with next term:
`(3sqrt7)/(sqrt21) = sqrt3`
So the ratio is correct.
Step 3: Use the sum of n terms formula
For a geometric series:
`S_n = a(r^n - 1)/(r - 1)`
Substitute a = √7 and r = √3:
`S_n = sqrt7((sqrt3)^n - 1)/(sqrt3 - 1)`
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