The second term of a GP is 4, the fifth term is 81. Find the seventh term
To find the seventh term of a geometric progression (GP), we need to determine the common ratio (r) first.
A geometric progression is defined by the formula:
\(a_n = a_1 \cdot r^{(n-1)}\)
Where:
- \(a_n\) is the nth term of the geometric progression
- \(a_1\) is the first term of the geometric progression
- \(r\) is the common ratio
- \(n\) is the position of the term in the sequence
Given that the second term, \(a_2\), is 4, we can substitute these values into the formula and solve for \(r\):
\(a_2 = a_1 \cdot r^{(2-1)}\)
4 = \(a_1 \cdot r\)
Similarly, the fifth term, \(a_5\), is 81:
\(a_5 = a_1 \cdot r^{(5-1)}\)
81 = \(a_1 \cdot r^4\)
Divide the equation \(a_5 = 81\) by \(a_2 = 4\) to eliminate \(a_1\):
\(\frac{{a_5}}{{a_2}} = \frac{{a_1 \cdot r^4}}{{a_1 \cdot r}}\)
\(\frac{{81}}{{4}} = r^3\)
Now, we can find the value of \(r\) by taking the cube root of 81/4:
\(r = \sqrt[3]{\frac{{81}}{{4}}}\)
Evaluate the cube root to find the value of \(r\):
\(r = \sqrt[3]{20.25} \approx 2.5\)
We now have the value of the common ratio, \(r\), which is 2.5. To find the seventh term of the geometric progression, substitute the values into the formula:
\(a_7 = a_1 \cdot r^{(7-1)}\)
\(a_7 = a_1 \cdot r^6\)
\(a_7 = a_1 \cdot (2.5)^6\)
We do not have the value of \(a_1\), the first term of the geometric progression, so it is not possible to find the seventh term without that information.
4 r^3 = 81 ... r = cuberoot (81 / 4)
81 r^2 = t7