
gesteigert
increased
Present to a higher or intensified degree; intensified.
Seit dem Vorfall hat sie ein gesteigertes Bedürfnis nach Sicherheit.
Since the incident, she has had an increased need for security.
increased
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increased
Present to a higher or intensified degree; intensified.
Seit dem Vorfall hat sie ein gesteigertes Bedürfnis nach Sicherheit.
Since the incident, she has had an increased need for security.

inflected for comparison
In grammar, the property of an adjective or adverb being in a comparative form or superlative form.
Das Wort „besser“ ist die gesteigerte Form von „gut“.
The word 'besser' (better) is the comparative form of 'gut' (good).
'Gesteigert' (increased) is the past participle of the verb 'steigern' (to increase, intensify). When it is used as an adjective, it keeps this meaning of 'increased' or 'intensified'.
Die Mannschaft zeigte in der zweiten Halbzeit eine gesteigerte Leistung.
The team showed an improved performance in the second half.
In everyday language, 'gesteigert' almost always means 'intensified' or 'increased'. The grammatical meaning is used only in specialist discussions about language.
Er hat ein gesteigertes Interesse an moderner Kunst.
He has an increased interest in modern art.
'Gesteigert' (increased) is often combined with abstract nouns to express a higher intensity. Typical examples are 'gesteigertes Interesse' (increased interest), 'gesteigerte Nachfrage' (increased demand), or 'gesteigertes Selbstvertrauen' (increased self-confidence).
Nach der Werbekampagne gab es eine gesteigerte Nachfrage nach dem Produkt.
After the advertising campaign, there was increased demand for the product.
This meaning is a technical term from grammar (linguistics). Outside of language classes or linguistics, this meaning is not used.
Im Grammatikbuch wird die gesteigerte Form von Adjektiven erklärt.
The grammar book explains the comparative and superlative forms of adjectives.
increased / increasingly
Vermehrt often refers to something occurring in greater numbers or more frequently than before. Gesteigert, by contrast, emphasizes that a degree or intensity has become higher. Choose vermehrt for more processes or cases and gesteigert for stronger activity or performance.

Seit der Werbung zeigt der Verein eine gesteigerte Aktivität bei der Aufnahme neuer Mitglieder.
Since the advertising campaign, the organization has shown increased activity in admitting new members.
Seit der Werbung erhält der Verein vermehrt Anfragen von neuen Mitgliedern.
Since the advertising campaign, the organization has been receiving more inquiries from new members.