#JavaScript (ES6), 66 bytes

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

[Try it online!](https://tio.run/##FY5BboMwFET3PYV3tuOp429wgEYmq54iioRFgSag7yqpcn0SRprF0@hJc0vP9Ojv17//T84/w7qOUTHmyDtjeO8xxRzbOQ7PtKiztXY2Ul7sLV9ZZa11bKdu17XNaVQOU2c6/TWvx7NHgRIBB1So0YAciEAeVIBKUAAdQBWoBjXwDn5bixKFcwHBuQpVqMtN3HL5sGO@f6f@V7GIregzP/Iy2CVPSiYlhRH8rtQiig1Gxe9v6ws "JavaScript (Node.js) – Try It Online")