Geometric Progression

A sequence $a_1, a_2, a_3, \ldots$ (with every term nonzero) is called a Geometric Progression (G.P.) if the ratio of every pair of consecutive terms is the same constant, called the common ratio, usually denoted $r$:

If the first term is $a$, the G.P. is written out as

so that each term is obtained from the previous one by multiplying by $r$.

The $n$th (general) term. Reading off the pattern, the first term needs $r$ multiplied in $0$ times, the second needs it $1$ time, and in general the $n$th term needs $r$ multiplied in $(n-1)$ times:

As with the A.P. formula, this follows by induction: true for $n=1$ (giving $a_1=a$), and if $a_n=ar^{n-1}$ then $a_{n+1}=a_n\cdot r = ar^{n-1}\cdot r = ar^n$, matching the formula at $n+1$.

Sum of the first $n$ terms — derivation by "multiply and subtract". Let

Multiply every term by $r$:

Now subtract the second equation from the first. Every middle term — $ar, ar^2, \ldots, ar^{n-1}$ — appears in both expressions and cancels exactly, leaving only the first term of the first line and the last term of the second line:

Equivalently, multiplying numerator and denominator by $-1$, this is often written $S_n=\dfrac{a(r^n-1)}{r-1}$ — both forms are identical and either may be used; the second is usually more convenient when $r>1$, since it avoids a negative numerator and denominator both. …