Page:Calculus Made Easy.pdf/158

 the whole time during which the value is growing. It is divided into $$10$$ periods, in each of which there is an equal step up. Here $$\dfrac{dy}{dx}$$ is a constant; and if each step up is $$\tfrac{1}{10}$$ of the original $$OP$$, then, by $$10$$ such steps, the height is doubled. If we had taken $$20$$ steps,



each of half the height shown, at the end the height would still be just doubled. Or $$n$$ such steps, each of $$\dfrac{1}{n}$$ of the original height $$OP$$, would suffice to double the height. This is the case of simple interest. Here is $$1$$ growing till it becomes $$2$$.

In Fig. 37, we have the corresponding illustration of the geometrical progression. Each of the successive ordinates is to be $$1+\dfrac{1}{n}$$, that is, $$\dfrac{n+1}{n}$$ times as high as its predecessor. The steps up are not equal, because each step up is now $$\dfrac{1}{n}$$ of the ordinate at that part of the curve. If we had literally $$10$$ steps, with $$\left(1+\tfrac{1}{10}\right)$$ for the multiplying factor, the final total would be