calculate multiplication factor to go from 1 to 100 in y attempts

Question

I have totalnumberofdays and percentage on day 1. I have to incrementally grow the currentpercentage every day so that it reaches 100 percent on day n

For ex:

If totalnumberofdays = 7, percentage on day 1 = 1

currentpercentage would be equal to percentage on day 1 the first day and then needs to grow incrementally.

Day 1   1 percent
Day 2   2 percent
Day 3   5 percent
Day 4   10 percent
Day 5   20 percent
Day 6   50 percent
Day 7   100 percent

How should currentpercentage be calculated for each day.

It is not clear what "grows incrementally" means. Can the percentage just grow by one every day, and then jump to $100$ on the last day? — Gerry Myerson, Jan 08 '20 at 23:39
@GerryMyerson: No, it should be uniformly and consistently growing but the percentage for a later day would always be greater than the prior day — systemdebt, Jan 08 '20 at 23:39
Notice that what you may be calling the "multiplication factor" in your example is two the first day but two-and-a-half the next day, so it isn't constant. How variable do you allow it to be? — Gerry Myerson, Jan 08 '20 at 23:40
Linear growth or exponential ? ... or something else ? ... Your example is neither. — Donald Splutterwit, Jan 08 '20 at 23:40
So, Simran, what do you mean by "uniformly"? As noted, your example is not uniform. — Gerry Myerson, Jan 08 '20 at 23:41
@DonaldSplutterwit Exponential seems do it going by the example — systemdebt, Jan 08 '20 at 23:47
It does not have to be exact. Just that it should reach 100% on day n. and grow each day. — systemdebt, Jan 08 '20 at 23:49
Let $N$ be the total number of days, $x$ be the day number and $y$ the percentage. Then a linear relationship is given by \begin{eqnarray} y=1+99 \times \frac{x-1}{N-1}. \end{eqnarray} — Donald Splutterwit, Jan 08 '20 at 23:51
For $y$ to increase exponentially \begin{eqnarray} y=\exp \left( \frac{(x-1) \ln(100)}{(N-1)} \right). \end{eqnarray} — Donald Splutterwit, Jan 08 '20 at 23:57
Does the percentage each day have to be a whole number, as in your example? — Gerry Myerson, Jan 09 '20 at 03:35

score 1 · Accepted Answer · answered Jan 09 '20 at 22:08

The problem is stated confusingly because growth is often calculated as a percentage, but growth in percentage uses this term twice. Are you sure you are not confused with the terms?

Apart from that, this just looks like an exponential function:

Define $p_1$ as the percentage on day 1, and in general $p_i$ the percentage on day $i$. And a daily growth of (percentage) $q$

Then we have:

$$p_i = p_1 \cdot q ^ {i-1}$$

With formulas from your textbooks (or comments), you should be able to calculate $q$ from $p_1$, $n$, etc.

OP seems to be happy with this answer, even though it doesn't fit the example OP gives. — Gerry Myerson, Jan 10 '20 at 16:01

calculate multiplication factor to go from 1 to 100 in y attempts

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