Proposition IX.35 — If as many numbers as we please be in continued proportion, …

Proof

For

a, a r, a r^{2}, \dots, a r^{n}

: the excess of the last over the first is

a (r^{n} - 1)

, and the sum of the rest is

a (1 + r + r^{2} + \dots + r^{n - 1}) = a (r^{n} - 1) / (r - 1)

; the ratio of excess to sum equals

(r - 1) = (a r - a) / a

, the excess of the second over the first divided by the first.

Knowledge graph · drag to pan, scroll to zoom, click a node to navigate

Full neighborhood

Depends on (3)

VII.11Proposition VII.11If, as whole is to whole, so is a number subtracted to a number subtracted, the remainder will also be to the remainder…
VII.19Proposition VII.19If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced…
VIII.1Proposition VIII.1If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the…

Required by (dependents) (1)

IX.36Proposition IX.36If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of…

Discussion

No replications, contradictions, or comments registered yet for this claim.

Replicate or annotate this claim

Replicate to register a fresh attempt; contradict, extend, or comment otherwise. Authors can post a claim-retraction with the reason taxonomy from RRP-0020.