#JavaScript (ES6), 68 bytes

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    f=(n,k=n*++n/2,g=o=>eval([...k+''].join(o)))=>(k=g`*`)>9?f(0,g`+`):k

[Try it online!][TIO-jgrzivui]

[TIO-jgrzivui]: https://tio.run/##FY7BjoIwGAbvPkVvbe23tX@hApqyJ5/CmNCwwCrk70aNr8/qYQ6TucwtvdKjv1//nl@cf4Z1HaNizJG3xvDOY4o5tsMrLepsrZ2NlBd7y1dWWWsdWzXHqdt2um2@R@UwdabTh3k9nj0KlAjYo0KNBuRABPKgAlSCAmgPqkA1qIF38J9alCicCwjOVahCXTaXjR3z/ZT6X8UitqLP/MjLYJc8KZmUFEbwG6lFFB8ZFb/H1n8 "JavaScript (Node.js) – Try It Online"