If |a_k|

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Less than one

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(D) We have $|\lambda_1 a_1 + \lambda_2 a_2 + ... + \lambda_n a_n| \le |\lambda_1 a_1| + |\lambda_2 a_2| + ... + |\lambda_n a_n|$ by the triangle inequality.Since $\lambda_k \ge 0$,this equals $\lambda_1 |a_1| + \lambda_2 |a_2| + ... + \lambda_n |a_n|$.Given $|a_k| Summing these inequalities,we get $\sum_{k=1}^n \lambda_k |a_k| Since $\sum_{k=1}^n \lambda_k = 1$,it follows that $|\lambda_1 a_1 + \lambda_2 a_2 + ... + \lambda_n a_n| < 1$.

If $|a_k| < 1, \lambda_k \ge 0$ for $k = 1, 2, ..., n$ and $\lambda_1 + \lambda_2 + ... + \lambda_n = 1$,then the value of $|\lambda_1 a_1 + \lambda_2 a_2 + ... + \lambda_n a_n|$ is

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