Question

in the diagram below, the total number of circles t when there are n rows is given by t = 0.5n² + 0.5n.

does the formula hold for the four figures shown?
∘ yes
∘ no

determine the number of rows when the total number of circles is 36.
______ rows

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Explanation:

First Sub - Question (Does the formula hold?):

Step 1: Check for \(n = 1\)

The formula is \(T=0.5n^{2}+0.5n\). When \(n = 1\), \(T=0.5\times1^{2}+0.5\times1=0.5 + 0.5=1\). The first figure has 1 circle. So it holds for \(n = 1\).

Step 2: Check for \(n = 2\)

When \(n = 2\), \(T=0.5\times2^{2}+0.5\times2=0.5\times4 + 1=2 + 1 = 3\). The second figure has 3 circles. So it holds for \(n = 2\).

Step 3: Check for \(n = 3\)

When \(n = 3\), \(T=0.5\times3^{2}+0.5\times3=0.5\times9+1.5 = 4.5+1.5=6\). The third figure has 6 circles. So it holds for \(n = 3\).

Step 4: Check for \(n = 4\)

When \(n = 4\), \(T=0.5\times4^{2}+0.5\times4=0.5\times16 + 2=8 + 2=10\). The fourth figure has 10 circles. So it holds for \(n = 4\).

Step 1: Set up the equation

We know that \(T = 0.5n^{2}+0.5n\) and \(T = 36\). So we set up the equation \(0.5n^{2}+0.5n=36\). Multiply both sides of the equation by 2 to get rid of the decimals: \(n^{2}+n = 72\). Then rewrite it as a quadratic equation: \(n^{2}+n - 72=0\).

Step 2: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 1\), \(b = 1\), \(c=-72\)), we can use the quadratic formula \(n=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) or factor the equation. Let's factor \(n^{2}+n - 72\). We need two numbers that multiply to \(- 72\) and add up to \(1\). The numbers are \(9\) and \(-8\). So \(n^{2}+n - 72=(n + 9)(n - 8)=0\).

Step 3: Find the value of \(n\)

Setting each factor equal to zero: \(n+9 = 0\) gives \(n=-9\) (not valid since the number of rows can't be negative) and \(n - 8=0\) gives \(n = 8\).

Answer:

Yes

Second Sub - Question (Number of rows when \(T = 36\)):