Question

a company launches its new website and tracks the number of visits to the site. predict the number of visits on the 10th day. day number of visits 1 12 2 18 3 26 4 41 5 61

Explanation:

Step1: Find the differences between consecutive terms

The differences between the number of visits are:
\(18 - 12=6\), \(26 - 18 = 8\), \(41-26 = 15\), \(61 - 41=20\)

Step2: Find the second - level differences

The differences of the first - level differences are:
\(8 - 6=2\), \(15 - 8 = 7\), \(20 - 15=5\)
Since the differences are not constant, we can try to fit a quadratic function \(y=ax^{2}+bx + c\)
Substitute \(x = 1,y = 12\); \(x = 2,y = 18\); \(x=3,y = 26\)
For \(x = 1,y=12\): \(a\times1^{2}+b\times1 + c=12\), i.e., \(a + b + c=12\)
For \(x = 2,y = 18\): \(a\times2^{2}+b\times2 + c=18\), i.e., \(4a+2b + c=18\)
For \(x = 3,y = 26\): \(a\times3^{2}+b\times3 + c=26\), i.e., \(9a+3b + c=26\)
Subtract the first equation from the second: \((4a + 2b + c)-(a + b + c)=18 - 12\)
\(3a + b=6\)
Subtract the second equation from the third: \((9a+3b + c)-(4a + 2b + c)=26 - 18\)
\(5a + b=8\)
Subtract \(3a + b=6\) from \(5a + b=8\): \((5a + b)-(3a + b)=8 - 6\)
\(2a=2\), so \(a = 1\)
Substitute \(a = 1\) into \(3a + b=6\), we get \(3\times1+b=6\), so \(b = 3\)
Substitute \(a = 1,b = 3\) into \(a + b + c=12\), we get \(1+3 + c=12\), so \(c = 8\)
The quadratic function is \(y=x^{2}+3x + 8\)

Step3: Predict the number of visits on the 10th day

When \(x = 10\), \(y=10^{2}+3\times10 + 8\)
\(y=100+30 + 8\)
\(y = 138\)

Answer:

138