Question

fill in the arithmetic means: 12. table with two columns: first column has entries x, 1, 2, 3, 4; second column has entries y, 18, (blank), (blank), -3

Explanation:

Step1: Identify the arithmetic sequence

We have an arithmetic sequence where when \( x = 1 \), \( y = 18 \); when \( x = 4 \), \( y=- 3\). Let the common difference be \( d \). The number of terms between \( x = 1 \) and \( x = 4 \) is \( 4 - 1=3\) intervals. So the change in \( y \) is \( - 3-18=-21\) over 3 intervals.

Step2: Calculate the common difference \( d \)

The common difference \( d=\frac{-21}{3}=- 7\)

Step3: Find \( y \) when \( x = 2 \)

When \( x = 2 \), it is 1 interval away from \( x = 1 \). So \( y=18 + (-7)=11\)

Step4: Find \( y \) when \( x = 3 \)

When \( x = 3 \), it is 2 intervals away from \( x = 1 \). So \( y = 18+2\times(-7)=18 - 14 = 4\)

Step5: Find \( y \) when \( x \) (first term)

Let the first term (when \( x\) is the first term, let's say \( x = 0\) for the first term, but actually we can find the general formula \(y=a+(x - 1)d\), when \( x = 1\), \(y = 18=a+(1 - 1)d\), so \(a = 18\). So the formula is \(y=18+(x - 1)\times(-7)=18-7x + 7=25-7x\)
When \( x\) is the first term (let's assume the first \( x\) is 0, but from the table, the first \( x\) is given as the first row, we can also see the sequence for \( x = 1,y = 18\); \(x = 2,y = 11\); \(x = 3,y = 4\); \(x = 4,y=-3\)

Answer:

When \( x = 2 \), \( y = 11 \); when \( x = 3 \), \( y = 4 \); the general formula for \( y\) in terms of \( x\) is \( y=25 - 7x\) (and the first term when \( x\) is the starting value, if we consider the first \( x\) as the first term, we can find it too, but from the table, the values are:

\( x \)	\( y \)

| (first \( x \), let's say if we consider the first term before \( x = 1\), but from the given table, the rows are \( x\): (let's assume the first \( x\) is the one before 1, but the given \( x\) values are 1,2,3,4. So the \( y\) values are: for \( x = 1\), \( y = 18\); \( x = 2\), \( y = 11\); \( x = 3\), \( y = 4\); \( x = 4\), \( y=-3\))

So filling the table:

For \( x = 2 \), \( y = 11 \); for \( x = 3 \), \( y = 4 \)