Is the following true? 

Let $a$ be a positive integer and let $t_n$ be a sequence of numbers. We define the binomial mean of $t_n$

$$
\beta_{t_n,a} =  \frac{1}{2^n t_n}\sum_{r^a \le n} \binom{n}{r^a}t_r
$$

> **Claim**: If the binomial mean $\beta_{t_n,a}$ approaches a finite positive constant and $f$ is continious on the interval $[0,1]$ then, 
$$
\frac{1}{2^n}\sum_{r^a \le n} \binom{n}{r^a}f\Big(\frac{t_r}{t_n}\Big) 
\sim f(\beta_{t_n,a}) $$

**Note**: The case $a = 1, t_n = n$ is true and is a special case of the approximation of $f$ using [Bernstein polynomials][1] as mentioned in the answer to the related question [Binomial analogue of Riemann sum for definite integral][2]

**Example**: Let $a = 2, t_n = p_n$ be the $n$-th prime and $f(x) = \log(x)$ for $n = 10^6$, LHS =$-0.7426766$, RHS =$-0.7426760$, and the error is $< 5.65 \times 10^{-7}$

**Note**: This question was posted in [MSE last month][3]. It got upvotes but no answer hence posting it in MO


  [1]: https://en.wikipedia.org/wiki/Bernstein_polynomial#Approximating_continuous_functions
  [2]: https://math.stackexchange.com/questions/3305329/binomial-analogue-of-riemann-sum-for-definite-integral
  [3]: https://math.stackexchange.com/questions/3306443/weighted-mean-of-a-function-with-binomial-coefficients-as-weights