1.3 Geometric sequences and seriesIB Maths: Analysis and Approaches SL: Revision notes

In a geometric sequence each term is the previous term multiplied by the same number, the common ratio $r = \frac{u_{n+1}}{u_n}.$ The $n$th term (in the formula booklet) is $u_n = u_1 r^{\,n-1}.$ For $5, 10, 20, \dots$: $u_1 = 5$, $r = 2$, so $u_8 = 5\times2^{7} = 640$.

The ratio can be negative (terms alternate in sign) or a fraction (terms shrink towards 0).

The sum of the first $n$ terms (in the booklet) is $S_n = \frac{u_1(r^{n}-1)}{r-1} = \frac{u_1(1-r^{n})}{1-r}, \quad r\ne1.$ Use whichever form keeps the denominator positive. For 8 weeks of cases starting at 5 and doubling: $S_8 = \frac{5(2^{8}-1)}{1} = 1275$.

With a negative ratio, bracket it carefully: $(-2)^{7} = -128$, not $128$.

$\sum_{k=1}^{n} u_1 r^{k-1} = u_1 + u_1 r + \dots + u_1 r^{n-1}.$ Recognise a geometric series in sigma notation by an exponent containing $k$, e.g. $\sum_{k=1}^{6} 3(2)^{k}$ has $u_1 = 6$ (put $k = 1$), $r = 2$ and 6 terms.

If you use technology to evaluate a sum, you must still be able to identify the first term and the common ratio.

A constant percentage change gives a geometric sequence:

• increase by $p\%$: multiplier $r = 1 + \frac{p}{100}$ (salary rising 4%: $r = 1.04$);
• decrease by $p\%$: multiplier $r = 1 - \frac{p}{100}$ (population falling 3%: $r = 0.97$).

Typical contexts are the spread of disease, salary increases or decreases, and population growth. Be careful whether $n$ counts terms (salary in year 10 is $u_{10} = u_1 r^{9}$) or years after a start (population after 10 years is $P_0 r^{10}$).

To find when a term passes a value, solve an inequality like $5\times2^{n-1} > 10\,000$. At SL you can use technology (a table or solver on the GDC) — laws of logarithms come later. Always check the integers either side and answer in context.

Comparing two geometric models (one growing, one decaying) is the same: solve $12\,000(1.05)^{n} > 25\,000(0.97)^{n}$ with the GDC ($n > 9.26$) and conclude "after 10 complete years". Comment on whether a constant percentage rate is realistic long term.