The third term of a G.P. is 360 and the sixth term is 1215. Find the:
In a geometric progression (G.P.), any term \(a_n\) can be expressed as \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio. We are given the third term \(a_3 = 360\) and the sixth term \(a_6 = 1215\). Using the formula, we have two equations:
1) \(a_1 \cdot r^2 = 360\)
2) \(a_1 \cdot r^5 = 1215\)
To find the common ratio \(r\), we divide the second equation by the first:
\[\frac{a_1 \cdot r^5}{a_1 \cdot r^2} = \frac{1215}{360}\]
Simplifying the left side, we get \(r^3\):
\[r^3 = \frac{1215}{360} = 3.375\]
Taking the cube root of both sides:
\[r = \sqrt[3]{3.375} = 1.5\]
Thus, the common ratio \(r\) is \(1.5\) (or \(\frac{3}{2}\)).
Now that we have \(r = 1.5\), we substitute it back into the equation for the third term to find the first term \(a_1\):
\[a_1 \cdot (1.5)^2 = 360\]
\[a_1 \cdot 2.25 = 360\]
Dividing by \(2.25\):
\[a_1 = \frac{360}{2.25} = 160\]
Thus, the first term \(a_1\) is \(160\).
The sum of the first \(n\) terms of a geometric progression is given by \(S_n = a_1 \cdot \frac{r^n - 1}{r - 1}\) for \(r
eq 1\). For \(n = 4\), \(a_1 = 160\), and \(r = 1.5\):
\[S_4 = 160 \cdot \frac{(1.5)^4 - 1}{1.5 - 1}\]
First, calculate \((1.5)^4\):
\[(1.5)^4 = 5.0625\]
Now substitute:
\[S_4 = 160 \cdot \frac{5.0625 - 1}{0.5} = 160 \cdot \frac{4.0625}{0.5} = 160 \cdot 8.125 = 1300\]
Thus, the sum of the first four terms \(S_4\) is \(1300\).
Geometric progressions are frequently used in finance to model compound interest. For example, if you invest an amount of money that grows at a fixed percentage rate every year, the value of your investment forms a geometric progression. By knowing the value of your investment at two different points in time, you can determine the interest rate (common ratio) and the initial amount invested, which helps in predicting the total value of your savings over a specific period.