1.2 Arithmetic sequences and seriesIB Maths: Applications and Interpretation HL: Revision notes

In an arithmetic sequence each term is found by adding a fixed number, the common difference $d$, to the previous term: $u_{n+1}=u_n+d$. The $n$th term is $u_n=u_1+(n-1)d.$ Example: $7,11,15,\ldots$ has $u_1=7$ and $d=4$, so $u_{20}=7+19\times4=83$. To find which term equals 135, solve $7+4(n-1)=135$, giving $n=33$. Given two terms, subtract: $u_8-u_3=5d$. If $u_3=11$ and $u_8=31$ then $d=4$ and $u_1=11-2(4)=3$.

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: for $u_1=7$, $d=4$ and $n=20$, $S_{20}=\frac{20}{2}\left(7+83\right)=900$. If you know the sum and need $n$, form an equation in $n$ and solve it on your GDC. Reject a negative or non-integer value of $n$.

$\sum_{k=1}^{n}u_k$ means $u_1+u_2+\cdots+u_n$: the sum of the terms from $k=1$ up to $k=n$. Example: $\sum_{k=1}^{12}(3k+2)$ is an arithmetic series with $u_1=5$, $u_{12}=38$ and 12 terms, so its sum is $\frac{12}{2}(5+38)=258$. The number of terms in $\sum_{k=a}^{b}$ is $b-a+1$.

A spreadsheet or GDC can list the terms of a sequence and their sums. Type the first term, then a rule such as 'previous term + $d$', and fill down. If you use technology in an exam you must still identify the first term $u_1$ and the common difference $d$ and show them in your working. Your GDC can also solve equations such as $\frac n2(2u_1+(n-1)d)=1275$ or find the first $n$ for which a term or sum passes a target. Check the answer in context.

Simple interest gives an arithmetic sequence: the interest each year is the same fixed amount. For 2500 EUR at 4% simple interest per year, the value after $t$ years is $2500+100t$, so after 6 years it is 3100 EUR. Real data are rarely perfectly arithmetic. Estimate $d$ from the data, for example $d\approx\frac{u_5-u_1}{4}$, and build a model $u_n=u_1+(n-1)d$. Use the model to predict, and always comment on reliability: extrapolating far beyond the data is risky because the real trend may change.