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The calculation below is from a NBER paper.

If $Y_t$ grows at the constant rate $g=r$, then $$ \triangle G_t+\frac{\triangle G_{t+1}}{1+r}+\frac{\triangle G_{t+2}}{(1+r)^2}+\dots$$ can be rewritten as $$ Y_t \left(\triangle \frac{G_t}{Y_t}+\triangle \frac{G_{t+1}}{Y_{t+1}}+\triangle \frac{G_{t+2}}{Y_{t+2}}+\dots\right) $$ I think this calculation is from $Y_{t+1}=Y_t (1+r)$ and $\frac{\triangle G_t}{Y_t}=\triangle \left(\frac{G_t}{Y_t}\right)$.
How $\frac{\triangle G_t}{Y_t}=\triangle \left(\frac{G_t}{Y_t}\right)$ relationship can be established?
Or are there any other missing clues I didn't noticed?

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How $\frac{\triangle G_t}{Y_t}=\triangle \left(\frac{G_t}{Y_t}\right)$ relationship can be established?

If $\Delta G_t={G_t-G_{t-1}}$ and $Y_t$ is fixed:

$$ \frac{\Delta G_t}{Y_t}=\frac {G_t-G_{t-1}}{Y_t}=\frac {G_t}{Y_t}-\frac {G_{t-1}}{Y_t}=\Delta \left(\frac{G_t}{Y_t}\right) $$.

It is just a different way of writing the same thing.

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