7. Given
$$
(1+x)^{50}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{50} x^{50} \text {. }
$$

then $a_{1}+2 a_{2}+\cdots+25 a_{25}=$ $\qquad$
$7.50 \times 2^{48}$.
From the expanded form, it is easy to know
$$
\begin{array}{l}
a_{1}+2 a_{2}+\cdots+25 a_{25} \\
=\mathrm{C}_{50}^{1}+2 \mathrm{C}_{50}^{2}+\cdots+25 \mathrm{C}_{50}^{25} .
\end{array}
$$

By $k \mathrm{C}_{50}^{k}=50 \mathrm{C}_{49}^{k-1}$, we know
$$
\begin{array}{l}
\mathrm{C}_{50}^{1}+2 \mathrm{C}_{50}^{2}+\cdots+25 \mathrm{C}_{50}^{25} \\
=50\left(\mathrm{C}_{49}^{0}+\mathrm{C}_{49}^{1}+\cdots+\mathrm{C}_{49}^{24}\right) \\
=50 \times \frac{1}{2} \times 2^{49}=50 \times 2^{48} .
\end{array}
$$