Let’s find the sum of a general geometric series.
$$
\begin{array}{rc}
S_n=&u_1&+&u_1r&+&u_1r^2&+&…&+&u_1r^{n-2}&+&u_1r^{n-1}
\end{array}
$$
If we multiply this series $S_n$ by $r$, we get
$$
\begin{array}{rc}
rS_n=r(u_1&+&u_1r&+&u_1r^2&+&…&+&u_1r^{n-2}&+&u_1r^{n-1})
\end{array}
$$
If we multiply $r$ by $u_1$, it becomes
If we multiply $r$ by $u_1r$, it becomes:
*write your answer in the form $u_1r^\square$
If we multiply $r$ by $u_1r^2$, it becomes
*write your answer in the form $u_1r^\square$
If we multiply $r$ by $u_1r^{n-2}$, it becomes
*write your answer in the form $u_1r^\square$
If we multiply $r$ by $u_1r^{n-1}$, it becomes
*write your answer in the form $u_1r^\square$
If we subtract $S_n$ from $rS_n$, we get
$$
\begin{array}{rc}
\color{red}{rS_n}=&&+&u_1r&+&u_1r^2&+&…&+&u_1r^{n-1}&+&u_1r^n\\
\color{red}{S_n}=&u_1&+&u_1r&+&u_1r^2&+&…&+&u_1r^{n-1}&\\
\hline
\fbox{$\color{red}{rS_n}-\color{red}{S_n}$}=\\
\end{array}
$$
Factorize $S_n$ from $\color{red}{rS_n}-\color{red}{S_n}$.
Find the sum of $3+6+12+24+48+\cdots$ to $13$ terms.
Find the sum of $4+12+36+108+\;\cdots\;+8748$.
How many terms are needed for the sum of the geometric series $4 + 12 + 36 + 108 + \cdots$ to exceed $100 000$?
Find the sum of the geometric series that has sixth term $6250$, ninth term $50$, last term $2$.
Consider a geometric series where the sum of the first two terms is $12$ and the sum of the first four terms is $60$. One sequence is $4, 8, 16, 32,\;\ldots$
Write down the first four terms of another sequence.
The sum of the first $n$ terms of a geometric series is given by $S_n = 5^n – 1$.
Find the sixth term of this series.
$u_{6}=$
The sum of the first two terms of a geometric series is $4$ and the sum of the first three terms is $-14$. Find the two possible values for the first term.
Separate your answers with a comma
$u_1=$
Find $n$ given $\displaystyle \sum_{k=1}^n2\times5^{k-1}=39062$
$n=$
A rope of length $6$m is cut into three pieces whose lengths form a geometric sequence.
The longest piece is four times the shortest piece. Find the length of the shortest piece.
Give your answer as a fraction
Josh puts £1000 into an account which earns 10% p.a. interest paid annually. At the end of each year, Josh puts in another £1000 in the account.
Find the value of the account after 6 years. Round your answer to the nearest £