Question Type 5: Finding n or the sum of first n terms when r = 1 Exercises

Show that for any constant sequence (geometric sequence with $r=1$) the sum of the first $2n$ terms is twice the sum of the first $n$ terms, i.e., $S_{2n}=2S_n$.

If $u_1=x$ and $r=1$ in a geometric series, and the sum of the first $n$ terms is $5x$, find $n$.

Derive a formula for the sum $S_n$ of the first $n$ terms of a geometric series with common ratio $r=1$ and first term $u_1=9$.

Calculate the sum of the first 15 terms of a geometric series with $u_1=9$ and $r=1$.

Calculate the sum of the first 50 terms of a geometric series with $u_1=12$ and $r=1$.

A machine produces 8 widgets per day for $n$ consecutive days, yielding a total of 560 widgets. Assuming this forms a geometric series with $r=1$, find $n$.

In an experiment, a constant dosage is given each day for 7 days, and the total dosage is 210 mg. Model this as a geometric series with $r=1$. Find the daily dosage.

For a geometric series with $u_1=a$ and $r=1$, where $a \neq 0$, the sum of the first $n$ terms is $100a$. Find $n$.

Find the sum of the first 20 terms of a geometric sequence with $u_1=5$ and $r=1$.

The sum of the first $n$ terms of a geometric series with $u_1=9$ and $r=1$ is 180. Determine $n$.

Express the sum $S_n$ of the first $n$ terms of a geometric series with first term $u_1$ and common ratio $r=1$ in terms of $u_1$ and $n$.

Given a geometric series with $u_1=7$, $r=1$, and sum of the first $n$ terms $S_n=210$, find $n$.