Say you have 12 K active customers every month in your platform in August 2016, how can you determine at what rate do you need to grow every month to achieve a certain total (say, 1.5 million) by next year, August 2017. Further, how can you generate different growth scenarios? Say, grow aggressively during the first quarter and then plateau. Are there any time series or optimization tools related to this kind of problem?
How can you determine the growth rate you need to achieve a certain customer base?
time series
A growth rate is an exponential growth, you multiply your userbase by a certain fraction, which is above 1 in case of growth, else it would be decline. This effect compounds because the amount of users is bigger while being multiplied by the same multiplier. Let's call this multiplier a. We apply this 12 times, once for every month. We are interested in:$$12000 \cdot a^{12} = 1500000$$This equals:$$a^{12} = 125$$We can take the 12th root on both sides:$$a = 1.495$$This means you would need to grow approximately 50% every month.For growing faster first and then plateauing you can do similar equations, although a bit more difficult. You have to express the growth at the first level in terms of the other level because else there are two unknowns which make for infinite solutions. Let's say the first 3 months the growth b is five times as high as the growth a last 9 months. This means $b = 5(a-1)+1$. This leads to the following equation.First we have three months of growth rate b, which means after three months our userbase is $12000\cdot b^3$ and after that we have nine months of growth a which leads to the final user base being $12000\cdot b^3\cdot a^9$. Since $b=5(a-1)+1$ this comes down to $12000\cdot (5(a-1)+1)^3\cdot a^9$, which according to Wolfram Alpha comes down to a being around 1.2785 which means b is around 2.3925. This means the first three months a growth of 139% and then nine months of 27.85% growth.