- #1

Aristarchus_

- 95

- 7

- Homework Statement
- The problem reads as follows

In a convergent geometric series with positive terms, the sum of the first and the third term is equal to the product of the first and the second term. (As this makes any sense...I can make the case where this is not true)

Find the exact values for the quotient ##k## in the series and the first term ##a_1## when the sum of the series is the smallest possible.

- Relevant Equations
- The solution is ##k = \sqrt{2}-1## and ##a_1 = 2 \cdot \sqrt{2}##

I do not have any reasonable attempts at this problem, as I am trying to figure out how one can get the correct answer when we are not given any values. Maybe if some of you sees a mistake here, that implies that the values from the previous example should be used...

##a_3 = a_1 \cdot k{2}## and ##a_2=a_1 \cdot k##

##a_1 + a_3 = a_1 \cdot (1+k^{2})## ##a_1 \cdot a_2 = a_1^{2} \cdot k##

from the problem, it then follows that ##a_1 \cdot (1+k^{2}) = a_1^{2} \cdot k##

But I do not see any connections yet...

What is meant by the smallest possible sum?

##a_3 = a_1 \cdot k{2}## and ##a_2=a_1 \cdot k##

##a_1 + a_3 = a_1 \cdot (1+k^{2})## ##a_1 \cdot a_2 = a_1^{2} \cdot k##

from the problem, it then follows that ##a_1 \cdot (1+k^{2}) = a_1^{2} \cdot k##

But I do not see any connections yet...

What is meant by the smallest possible sum?

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