Create an equation to solve the below sequences problem...

Question

Dr Vasee can make four robots a day. Each of the robots can also make 1 other robot per day, starting the next day. So on the first day, we have four robots. On the second day, we have four more Dr Vasee made, plus four more made by each of the previous day's robots. Thus we have a total of 12 robots = [ (4 X 2) + 4]. Thus the formula for total robots on each day is:

Total Robots = ( PreviousDayTotal X 2) + 4

I am looking for an equation to directly calculate the total number robots after x days. For eg, how many robots are there at the end of day 25?

Please help me with the equation. It would be great if you can also tell me how you arrived at this equation.

Answer 1

One formula which describes the total amount after $n$ days would be$$\sum_{i = 1}^n 2^{i+1} = \sum_{i = 0}^{n - 1} 2^{i+2} = 2^2 \sum_{i = 0}^{n - 1} 2^i = 4(2^n - 1).$$

How did we find that formula?

As you did, I computed the total amount for the first few days ($4$, $12$, $28$ etc.) and just looked at the numbers to figure out their relation and then you can prove that idea by induction here.

You can also get an idea how to arrive at the left-hand side. If you start with $$x_n = 2x_{n-1} + 4 = 2(2x_{n-2} + 4) + 4 = 2(2(2x_{n-3} +4 ) +4) + 4 =\; ...$$

The smallest summand is $4 = 2^2$ (on the very right end), the next one $2 \cdot 4 = 2^3$ and so on. Thus you can derive an easy formula.

Answer 2

Let ${a_n}$ be the sequence of the number of robot on the $n^{th}$ day. then, as you have calculated, $a_{n+1}=2a_n+4$ for $n\ge1$. By some pattern finding, we can find that the $a_n=4(2^n-1)$. To prove this, we will use mathematical induction.

When $n=1$, we have $4$ robots and $a_1=4(2^1-1)=4$.

The proposition is true for $n=1$.

Assume the proposition is true for $n=k$, that means, $$a_k=4(2^k-1)$$ When $n=k+1$, $$a_{k+1}=2a_k+4$$ $$=2(4(2^k-1))+4$$ $$=4(2(2^k-1)+1)$$ $$=4(2^{k+1}-1)$$ The proposition is true for $n=k+1$ if the proposition is true for $n=k$ is true. By the principle of Mathematical Induction, the above preposition is true for all n.