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1

Hint: It looks to me like a linear function (you could guess this by graphing points if you want). In that case, the formula will be $y=mx+b$ for some $m$ and some $b$. You could try plugging in values of $x$ and solving for $m$ and $b$. You might start by trying the case when $x=0$...

2

The sum of all digits of each row is always 20. Second row: $1+1+3+3+6+6=20$ Third row: $8+3+2+7=20$ Then: First row: $2 + 9 +a +b = 11+a+b = 20 \Rightarrow a+b = 9$ where $10a + b$ is the number to be placed in the position $(1,3)$. This is only possible if the number to be placed is $45$; in fact $a=4, b=5$ and $a+b=9$.

0

We know that the common ratio between terms is $-2$, we can see that from the series: $$S = 1 + (-2)1 + (-2)(-2) + (-2)4 + (-2)(-8) + ...$$ This means that the formula for the sum is: \begin{align*} S&= \frac{1 - (-2)^n}{1 - (-2)}\\ &= \frac{1 - (-2)^n}{3} \end{align*}

0

I don't know why it's important, but the third column is the first column divided by $200$. I figured it out from looking at the rows in red. I don't know why the numbers in green were missing, but it bothered me, so I fixed it. $$\begin{bmatrix} 59 & \color{green}{0.00} & \color{green}{0.295}\\ 148 & 0.60 & \color{green}{0.74}\\ ... 0 I found this guys:$$a(n) = \frac52(-3+(-1)^n+6n)$$Its on OEIS.org. ( OEIS A084957 - multiples of 5 whose GCD with 6 is 1.) 0 I would say :$$P_n=10(n+E(\frac{n+1}{2}))+5 $$Where E(x) is the greatest integer which is less or equal to x. 0 Hint: You can modify this sequence a bit to get yours. 0 To simplify it some we can "cheat" and do something like this: A[1] = 200 A[2..3] = 1/2 A[x-1] A[4..8] = -(31 x^4)/24+(281 x^3)/12-(3209 x^2)/24+(3259 x)/12-100 but plugging in 1 for sequence 4 (value 60), 2 for sequence 5 (value 75)...$$-\frac{31 x^4}{24} + \frac{281 x^3}{12} - \frac{3209 x^2}{24} + \frac{3259 x}{12} - 100 $$or (since the ... 6 For any finite sequence of real numbers (as in your example), there is a formula that exactly reproduces it, e.g. an interpolating polynomial. For your example sequence [200,100,50,60,75,39,15,35] (indexed from 1 to 8), the lowest degree interpolating polynomial is$$-\frac{163 x^7}{2520} + \frac{17 x^6}{9} - \frac{3947 x^5}{180} + \frac{9337 ...

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