Quick questions on Geometric sequences and series for HSC Maths Advanced financial modelling

A geometric sequence has a constant ratio $r$ between consecutive terms. With first term $a$,

The sum of the first $n$ terms is

If $|r| < 1$, then $r^n \to 0$ as $n \to \infty$, and the series converges to

A principal $P$ at compound rate $r$ per period produces the sequence of balances $P, P(1 + r), P(1 + r)^2, \dots$ with common ratio $1 + r$. The balance after $n$ periods is the $(n + 1)$th term, which gives the familiar $A = P(1 + r)^n$.

An asset depreciating at rate $d$ per period has values $V_0, V_0(1 - d), V_0(1 - d)^2, \dots$, a geometric sequence with ratio $1 - d$. The value after $n$ periods is $V_n = V_0 (1 - d)^n$. This is the "declining balance" method.

A series of equal payments made at regular intervals forms a geometric sum after each payment is moved to a common time using the compound interest factor. When the payments continue forever and the discount rate satisfies $|v| < 1$, the limiting sum gives the present value of a perpetuity.

Find $T_8$ and $S_8$ for the geometric sequence $3, 6, 12, 24, \dots$.

Find the limiting sum of $8 - 4 + 2 - 1 + \cdots$.

If you deposit $\$100$ at the end of each year into an account paying $5\%$ per annum, the balance just after the $n$th deposit is

A scholarship pays $\$5000$ at the end of each year forever, discounted at $4\%$ per annum. The present value is the limiting sum of $5000 v + 5000 v^2 + \cdots$ with $v = \frac{1}{1.04}$.

The first term $a$ is the first term of the sequence as written. In $T_n = a r^{n - 1}$, $a = T_1$, not $T_0$.

$T_n$ uses $r^{n - 1}$, not $r^n$. The sum $S_n$ uses $r^n$. The $n$th term of $P, P(1 + r), P(1 + r)^2, \dots$ is $P(1 + r)^{n - 1}$, but the compound interest balance after $n$ periods is $P(1 + r)^n$ because we count compounding events, not list positions.

The limiting sum formula needs $|r| < 1$ strictly. For $r = 1$ the sequence is constant and the partial sums diverge. For $|r| \ge 1$ but $r \neq 1$, the partial sums grow without bound or oscillate.

A $15\%$ depreciation per year means a multiplier of $0.85$ per year, not $0.15$. The new value is $0.85$ times the old.

Some questions list payments starting one period from now (an "ordinary annuity"), others start immediately (an "annuity due"). The first term and the number of compounded periods change accordingly.