find the sum of a gp 4+6+9...729/16
To find the sum of a geometric progression (GP), we can use the formula:
\[S_n = a \dfrac{r^n - 1}{r - 1}\]
where:
- \(S_n\) is the sum of the first \(n\) terms of the GP.
- \(a\) is the first term of the GP.
- \(r\) is the common ratio of the GP.
- \(n\) is the number of terms we want to sum.
In this case, the first term \(a = 4\), the common ratio \(r\) can be found by dividing the second term by the first term, i.e., \(r = 6/4 = 1.5\), and the last term can be calculated by the formula for the \(n\)th term of a GP:
\[a_n = a \cdot r^{(n-1)}\]
Given that the last term is 729/16, we can find the number of terms using the equation:
\[a_n = a \cdot r^{(n-1)}\]
\[\dfrac{729}{16} = 4 \times (1.5)^{n-1}\]
\[182.25 = 6^{n-1}\]
Solving for \(n\) gives us \(5\). Now, we can plug in these values into the sum formula:
\[S_5 = 4 \times \dfrac{1.5^5 - 1}{1.5 - 1}\]
\[S_5 = 4 \times \dfrac{57.375 - 1}{0.5}\]
\[S_5 = 4 \times \dfrac{56.375}{0.5}\]
\[S_5 = 4 \times 112.75\]
\[S_5 = 451\]
Therefore, the sum of the given geometric progression is 451.