The following pattern is given:
X=1,2,3,4,5
Y=3,6,11,18,27.
To find the equation let's check the first pair: x=1 and y=3.
If we power 1 to any number we obtain 1, but the result for y is 3. Then, we need to add 2 to obtain this result:
[tex]\begin{gathered} 3=1^1+a\text{ (the power doesn't matter in this case)} \\ 3-1=a \\ 2=a \end{gathered}[/tex]Now, we know the term we need to add is 2.
Let's analyze the second pair x=2 and y=6:
[tex]\begin{gathered} 6=2^b+2 \\ \text{Let's solve for b:} \\ 2^b=6-2 \\ 2^b=4 \\ \text{If we square the number 2, we obtain 2x2=4, then} \\ 2^2=4 \\ b=2 \end{gathered}[/tex]The equation would be then:
[tex]y=x^2+2[/tex]Let's check this when x=5:
[tex]\begin{gathered} y=5^2+2 \\ y=25+2 \\ y=27 \end{gathered}[/tex]This is correct, then we proved the equation is:
[tex]y=x^2+2[/tex]