The following gives a geometric sequence 2,p,q,250 find the value of p and Q
To find the values of p and q, we can use the formula for finding the terms of a geometric sequence:
\[a_n = a_1 \cdot r^{n-1}\]
In this case, we know that the first term, \(a_1\), is 2, and the third term, \(a_3\), is 250.
To find p, the second term, we can substitute n = 2 into the formula:
\[a_2 = a_1 \cdot r^{2-1}\]
\[p = 2 \cdot r\]
Similarly, to find q, we can substitute n = 3 into the formula:
\[a_3 = a_1 \cdot r^{3-1}\]
\[250 = 2 \cdot r^2\]
Now we have two equations:
\[p = 2r\]
\[250 = 2r^2\]
Substituting p = 2r into the second equation:
\[250 = p^2\]
Taking the square root of both sides:
\[\sqrt{250} = p\]
Simplifying the square root:
\[\sqrt{250} = \sqrt{25 \cdot 10} = \sqrt{25} \cdot \sqrt{10} = 5\sqrt{10}\]
Therefore, the value of p is \(5\sqrt{10}\).
Now we can substitute p = 5\(\sqrt{10}\) into the equation \(p = 2r\) to find q:
\[5\sqrt{10} = 2r\]
Solving for r:
\[r = \frac{5\sqrt{10}}{2}\]
To find q, the third term, we can substitute n = 3 into the formula:
\[q = a_1 \cdot r^{3-1}\]
\[q = 2 \cdot \left(\frac{5\sqrt{10}}{2}\right)^2\]
Simplifying:
\[q = 2 \cdot \frac{25 \cdot 10}{4} = 50 \cdot \frac{10}{4} = 125\]
Therefore, the value of q is 125.