Arithmetic and geometric sequencesIB MYP Maths Extended: Revision notes

In an arithmetic sequence you add the same number each time, the common difference $d$. Example: $7, 11, 15, 19,\dots$ has $d=4$. The $n$th term is $u_n=u_1+(n-1)d.$ For $u_1=7$ and $d=4$: $u_{20}=7+19\times4=83$. To find which term equals $103$, solve $7+(n-1)4=103$, so $n=25$. The difference can be negative, for a sequence that decreases.

An arithmetic series is the sum of the terms. The sum of the first $n$ terms is $S_n=\frac n2\left(2u_1+(n-1)d\right)=\frac n2\left(u_1+u_n\right).$ Example: $7+11+\dots$ to $20$ terms gives $S_{20}=\frac{20}{2}(14+76)=900$. The second form says: number of terms times the average of the first and last.

In a geometric sequence you multiply by the same number each time, the common ratio $r$. Example: $3, 6, 12, 24,\dots$ has $r=2$. Find $r$ by dividing a term by the one before it. The $n$th term is $u_n=u_1r^{n-1}.$ Example: $u_6=3\times2^5=96$. A ratio between $0$ and $1$ gives a sequence that shrinks, such as $80, 40, 20,\dots$ with $r=\frac12$.

The sum of the first $n$ terms of a geometric series with $r\ne1$ is $S_n=\frac{u_1\left(r^n-1\right)}{r-1}=\frac{u_1\left(1-r^n\right)}{1-r}.$ Use the first form when $r>1$ and the second when $r<1$, to keep the numbers positive. Example: $3+6+12+24+48+96=\frac{3(2^6-1)}{2-1}=189$.

A quantity that grows by a fixed percentage each period is a geometric sequence. For $4\%$ growth the multiplier is $1.04$; for a $20\%$ rise it is $1.2$; for a $10\%$ fall it is $0.9$. Compound interest on AED $2000$ at $4\%$ per year gives $2000\times1.04^n$ after $n$ years: $2000\times1.04^5=2433.31$. Simple interest is different: it adds the same amount each year, so it is arithmetic. To find when a target is reached, test values of $n$ (or use logarithms) and show the values either side of the target.

Decide first: is the difference between terms constant (arithmetic) or the ratio (geometric)? Check two pairs. Then pick the formula, work with the exact values and round only at the end. In context, give units, say what $n$ stands for and compare plans with numbers. Example: saving AED $50$ then $15$ more each month (arithmetic) gives AED $215$ in month $12$, but saving $20\%$ more each month (geometric) gives AED $371.50$, so a monthly limit could be broken.