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,

Fig. 36.

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